Fair Domination on Extended Supergrid Graphs: Complexity, Linear-Time Algorithms, and ILP Formulations ^†

Abstract

A dominating set of a graph is a subset of vertices such that every vertex is either contained in the set or adjacent to at least one vertex in it. A dominating set is called k-fair if each vertex not in this set is adjacent to exactly k vertices of the set. The domination and k-fair domination problems aim to find such sets of minimum cardinality. Both problems are NP-complete for general graphs, and the domination problem remains NP-complete on grid graphs, whereas the k-fair domination problem remains open on grid graphs. In this paper, we study the 1-fair and 2-fair domination problems on extended supergrid graphs, which generalize grid graphs and include both grid and supergrid graphs as subclasses. We prove that the 1-fair domination problem is NP-complete for these graph classes, even when restricted to planar graphs with maximum degree 4. On the positive side, for rectangular supergrid graphs, we present a linear-time algorithm for computing minimum 1-fair dominating sets. In addition, we formulate an integer linear programming (ILP) model to investigate the 1-fair and 2-fair dominations on small instances and introduce a restricted k-fair domination problem motivated by the experimental observations.

1. Introduction

Domination and its variants have been extensively studied in graph theory. The domination problem is known to be NP-complete for general graphs and remains NP-complete for planar graphs with maximum degree 3 [1], grid graphs [2], cubic planar graphs [3], chordal graphs [4], circle graphs [5], and many other graph classes. In contrast, the domination problem can be solved in polynomial time on several highly restricted graph classes, including trees [6], interval graphs [7], and cographs [8]. These sharp complexity boundaries have motivated further investigations into domination variants—including paired domination, restrained domination, and fair domination—on structured graph families. In particular, while domination is known to be NP-complete on grid graphs, the computational complexity of several domination variants on grid-like and supergrid-like graph classes remains open, providing strong motivation for recent research in this direction. Comprehensive surveys and references on domination and its variants can be found in the monographs by Haynes, Hedetniemi, and Slater [9,10], as well as in Johnson’s NP-completeness columns [11] and subsequent works that classify domination problems across numerous graph families.

Fair domination was introduced as a refinement of domination to impose local balance constraints [12]. The authors showed that the 1-fair domination problem coincides with the perfect domination problem. Nevertheless, this notion differs from efficient domination (also known as perfect codes), in which the closed neighborhoods of the dominating vertices form a partition of the vertex set; that is, every vertex is dominated by exactly one vertex of the dominating set.

Several studies have investigated fair domination in specific graph classes. Maravilla et al. [13] examined k-fair dominating sets in graphs obtained from operations such as join, corona, composition, and Cartesian product, providing bounds or exact values for their k-fair domination numbers. Hajian and Rad [14] determined an upper bound for the fair domination number of the cactus graphs and showed that this bound coincides with the 1-fair domination number for these graphs. In the same year, Jayasree and Radha [15] calculated the 1-fair domination number for several special graphs, including paths, cycles, Sierpiński graphs, and rectangular supergrid graphs. Hajian et al. [16] further established an upper bound for the fair domination number of outerplanar graphs. Sangeetha et al. [17] systematically calculated the fair domination number for several special graphs, including mill graphs, sunlet graphs, crown graphs, ladder graphs, prism graphs, gear graphs, web graphs, and helm graphs. More recently, Alikhani and Safazadeh [18] analyzed the number of fair dominating sets in several special graphs, such as complete bipartite graphs, cycles, paths, sociable friendship graphs, and triangular cactus graphs. Despite this growing body of work, the complexity of the 1-fair and 2-fair domination problems on grid graphs remains unresolved.

Supergrid graphs and their variants arise naturally in the study of grid-based network models and have attracted increasing attention in recent years. An extended supergrid graph generalizes a grid graph by allowing additional edges while preserving a grid-like structure, and it contains both grid graphs and supergrid graphs as special cases. Despite their structural regularity, domination-type problems on extended supergrid graphs remain far from fully understood, especially under fairness constraints. The supergrid graphs were first introduced in [19], where it was shown that the Hamiltonian cycle and path problems for these graphs are NP-complete. Supergrid graphs and related grid-like graph classes have also been considered in domination-type problems [20,21]. However, fair domination on extended supergrid graphs has not previously been investigated from a theoretic complexity perspective.

Our Contributions

In this paper, we investigate the 1-fair and 2-fair domination problems on extended supergrid graphs from a computational complexity perspective. Our main focus is on establishing hardness results, complemented by positive results for restricted graph classes. The contributions of this paper can be summarized as follows:

• We prove that the 1-fair domination problem is NP-complete on extended supergrid graphs, even when restricted to planar graphs with maximum degree 4. In [22], we claimed that the 2-fair domination problem on extended supergrid graphs is NP-complete; however, the proof contains a flaw. Therefore, its complexity remains open.
• We analyze the structural properties of rectangular supergrid graphs with respect to fair domination and present a linear-time algorithm to compute minimum 1-fair dominating sets. We further provide a linear-time algorithm for the 2-fair domination problem on small rectangular supergrid graphs. The computational complexity of the 2-fair domination problem on large rectangular supergrid graphs, however, remains open.
• We formulate an integer linear programming (ILP) model for computing optimal k-fair dominating sets and apply it to small supergrid graphs. The experimental results reveal that, due to the fairness constraints, many graphs admit the entire vertex set as the unique fair dominating set. Motivated by this observation, we introduce a variant called the restricted fair domination problem, whose computational complexity and potential applications merit further study.

These results reveal a sharp contrast between the intractable behavior of fair domination on general extended supergrid graphs and the tractable structure of rectangular subclasses, while the ILP-based study provides additional insight into structural irregularities and motivates the restricted fair domination model.

The remainder of this paper is organized as follows. Section 2 introduces the notation and preliminary concepts. Section 3 establishes the NP-completeness of the 1-fair domination problem on extended supergrid graphs. Section 4 investigates the 1-fair and 2-fair domination numbers of rectangular supergrid graphs. Section 5 presents the ILP model and introduces the restricted fair domination variant. Finally, Section 6 concludes the paper and discusses directions for future research.

2. Preliminaries and Definitions

All graphs considered in this paper are simple and undirected. For a graph G, let $V(G)$ and $E(G)$ denote its vertex set and edge set, respectively. For a vertex $v\in V(G)$, the open neighborhood of v is $N_G(v)=\{u\in V(G)\mid (u,v)\in E(G)\}$, and the closed neighborhood of v is $N_G\left[v\right]=N_G(v)\cup \left\{v\right\}$. The degree of v in G is $de{g}_G(v)=|N_G(v)|$. For a subset $S\subseteq V(G)$, the subgraph induced by S is denoted by $G\left[S\right]$, and we define $N_G(S)={\bigcup }_{v\in S}{N}_G(v)$ and $N_G\left[S\right]={\bigcup }_{v\in S}{N}_G\left[v\right]$. A path P in a graph G is a sequence $(v_1,v_2,\dots ,v_k)$ of adjacent vertices that begins at $v_1$ and ends at $v_k$, called a $(v_1,v_k)$-path. All vertices $v_1,v_2,\dots ,v_k$ are distinct, except in the special case where the path forms a cycle and $v_1=v_k$. When there is no ambiguity, a path on n vertices is denoted by $P_n$.

2.1. Supergrid-Related Graph Classes

An infinite two-dimensional integer supergrid $S^{\infty }$ is an infinite graph whose vertices are placed at all integer coordinate points in the two-dimensional Euclidean plane. Each vertex v is indicated by $(v_x,v_y)$, where $v_x$ and $v_y$ represent its x- and y-coordinates, respectively. In $S^{\infty }$, two vertices u and v are adjacent if and only if $|u_x-v_x|\le 1$ and $|u_y-v_y|\le 1$, where $|u_x-v_x|$ and $|u_y-v_y|$ denote the absolute differences between the x-coordinates and y-coordinates of the two vertices, respectively. Thus, the supergrid contains horizontal, vertical, and two types of diagonal edges (left-skewed and right-skewed). A horizontal (respectively, vertical) path in $S^{\infty }$ is a sequence of consecutive horizontal (respectively, vertical) edges. For example, Figure 1a depicts a partial segment of the infinite graph $S^{\infty }$.

Figure 1. Representative portions of infinite graphs. (a) A portion of the infinite two-dimensional integer supergrid $S^{\infty }$, whose vertices are located at integer coordinates $(x,y)$ and whose edges include horizontal, vertical, and diagonal adjacencies. The slanted edges represent the diagonal adjacencies. (b) A portion of the infinite two-dimensional integer grid $G^{\infty }$, obtained from $S^{\infty }$ by deleting all diagonal edges, so that adjacency occurs only between horizontally or vertically adjacent integer-coordinate vertices. The symbols “…” indicate that both graphs extend infinitely, and only representative coordinate labels are shown for clarity.

An infinite two-dimensional integer grid $G^{\infty }$ is obtained from $S^{\infty }$ by removing all diagonal edges. Two vertices w and z in $G^{\infty }$ are adjacent if and only if $|w_x-z_x|+|w_y-z_y|=1$. For example, Figure 1b depicts a partial segment of the infinite graph $G^{\infty }$.

A supergrid graph is any finite, connected vertex-induced subgraph of $S^{\infty }$. Equivalently, its vertices correspond to a finite subset of integer lattice points in ${\mathbb{Z}}^2$, and two vertices are adjacent if and only if their Euclidean distance is at most $\sqrt{2}$.

A grid graph is any finite, connected vertex-induced subgraph of $G^{\infty }$. Equivalently, its vertices correspond to a finite subset of integer lattice points in ${\mathbb{Z}}^2$, and two vertices are adjacent if and only if their Euclidean distance is one, that is, they differ by one in exactly one coordinate.

It should be noted that in grid graphs each vertex has a degree of at most 4, and every grid graph is both bipartite and planar [23]. However, in supergrid graphs, a vertex may have a degree up to 8, and such graphs are not necessarily bipartite or planar. This indicates that supergrid graphs have a more complex structure, which in turn leads to greater computational challenges when dealing with domination-related problems.

An extended supergrid graph is a finite, connected subgraph of $S^{\infty }$, while a grid graph (respectively, supergrid graph) is a finite, vertex-induced subgraph of $G^{\infty }$ (respectively, $S^{\infty }$). Note that an extended supergrid graph is not necessarily a vertex-induced subgraph of $S^{\infty }$. Thus, this class includes grid graphs and supergrid graphs as subclasses.

A rectangular supergrid graph (also called a King’s graph) is the vertex-induced subgraph of $S^{\infty }$ on the vertex set $\{(x,y)\in {\mathbb{Z}}^2\mid 1\le x\le n,1\le y\le m\}$. Equivalently, it is the strong product $P_m⊠P_n$ of the paths $P_m$ and $P_n$. Two distinct vertices u and v are adjacent if and only if $0\le |u_x-v_x|\le 1$ and $0\le |u_y-v_y|\le 1$, and $(u_x,u_y)\ne (v_x,v_y)$. Such graphs exhibit strong regularity properties that play a crucial role in the design of efficient algorithms for fair domination. A rectangular grid graph is the vertex-induced subgraph of $G^{\infty }$ on the same vertex set. Equivalently, it is the Cartesian product $P_m\square P_n$ of the paths $P_m$ and $P_n$, where two distinct vertices u and v are adjacent if and only if $|u_x-v_x|+|u_y-v_y|=1$.

Figure 2 presents examples of a grid graph, a rectangular grid graph, a supergrid graph, a rectangular supergrid graph, and an extended supergrid graph that is not an induced subgraph of $S^{\infty }$. In Figure 2, we assume that the vertex with coordinates $(1,1)$ is located at the upper-left corner, and the coordinates increase to the right and downward. For illustration, Figure 2a displays the coordinates of the vertices in the grid graph. Figure 3 illustrates the relationships among these graph classes.

Figure 2. Examples of graphs: (a) grid graph, (b) rectangular grid graph, (c) supergrid graph, (d) rectangular supergrid graph, and (e) extended supergrid graph that is neither a grid nor a supergrid graph. In (e), the green nodes indicate vertices that are not contained in the vertex-induced subgraph of $S^{\infty }$.

Figure 3. Inclusion relationships among grid graphs, supergrid graphs, and extended supergrid graphs, where $\mathcal{C}\to {\mathcal{C}}^{\prime }$ indicates that ${\mathcal{C}}^{\prime }$ is a subclass of $\mathcal{C}$.

2.2. Domination and Fair Domination

A dominating set of a graph G is a subset $D\subseteq V(G)$ such that $N_G\left[v\right]\cap D\ne \varnothing$ for every vertex $v\in V(G)$. For a vertex v and a dominating set D of G, v is said to be dominated by D. The domination number of G, denoted by $\gamma (G)$, is the minimum cardinality of a dominating set of G. The domination problem is to find a dominating set of size $\gamma (G)$.

Let k be a positive integer. A dominating set F of G is called a k-fair dominating set if every vertex $v\in V(G)\setminus F$ satisfies $|N_G(v)\cap F|=k$. The k-fair domination problem is to find a k-fair dominating set of minimum cardinality. Since $F=V(G)$ is a k-fair dominating set of G (because $V(G)\setminus F=\varnothing$), every graph admits a trivial k-fair dominating set. Let ${\gamma }_{k\mathrm{f}\mathrm{d}}(G)$ denote the minimum cardinality of a k-fair dominating set of G.

Clearly, if the maximum degree of the vertex of a graph G is $\Delta$, then the only $(\Delta +\tau )$-fair dominating set in G is $V(G)$, where $\tau \ge 1$. For example, in grid graphs (which have a maximum degree 4), the only 5-fair dominating set is the entire vertex set. Figure 4a depicts a minimum 1-fair dominating set of a grid graph, while Figure 4b shows a 1-fair dominating set that is not minimum, and Figure 4c shows a dominating set that is not 1-fair.

Figure 4. Examples illustrating fair domination. (a) A minimum 1-fair dominating set of a grid graph; (b) a 1-fair dominating set that is not minimum; and (c) a dominating set that is not 1-fair. Solid circles denote vertices in the (fair) dominating set, while hollow circles denote vertices outside the set.

3. NP-Completeness Results

In this section, we establish the computational hardness of fair domination on extended supergrid graphs. We prove that the 1-fair domination problem is NP-complete for grid and supergrid graphs. Moreover, the reduction extends to planar graphs with maximum degree 4. Chen et al. [20] showed that the domination and independent domination problems on supergrid graphs are NP-complete. Following a similar approach, we show that the 1-fair domination problem also remains NP-complete on this graph class and on grid graphs. Our proof is based on a polynomial time reduction from the domination problem on grid graphs, which was proved NP-complete by Clark et al. [2]. Specifically, we transform any instance of the domination problem on grid graphs into an equivalent instance of the 1-fair domination problem on grid and supergrid graphs.

Theorem 1

([2]). The domination problem on grid graphs is NP-complete.

We first prove that the 1-fair domination problem is NP-complete on grid graphs. Given a grid graph $G_g$, we construct another grid graph, denoted by $G_g^{1\mathrm{fd}}$. The key property of the construction is that $G_g$ has a dominating set D with $|D|\le k$ if and only if $G_g^{1\mathrm{fd}}$ has a 1-fair dominating set $D_g^{1\mathrm{fd}}$ with $|D_g^{1\mathrm{fd}}|\le k+2|E(G_g)|$.

Given an input grid graph $G_g$, we construct a new grid graph $G_g^{1\mathrm{fd}}$ as follows (see Algorithm 1):

Algorithm 1 Grid graph construction algorithm

Input: A grid graph $G_g$. Output: The constructed grid graph $G_g^{1\mathrm{fd}}$. Step 1: Construct the 5-subdivision graph $G_g^{\prime }$ of the input grid graph $G_g$. For each edge $(u,v)\in E(G_g)$, subdivide it by inserting four new vertices so that $(u,v)$ is replaced by a path of length 5 between u and v. Since $G_g$ is a grid graph, each subdivided edge forms a horizontal or vertical path. The resulting graph is the 5-subdivision of $G_g$, denoted by $G_g^{\prime }$. An example of $G_g^{\prime }$ is shown in Figure 5b for the grid graph $G_g$ in Figure 5a. Step 2: Introduce square tentacles. For each $(u,v)$-path in $G_g^{\prime }$ corresponding to an original edge $(u,v)\in E(G_g)$, we replace the path with a connected subgraph called a $(u,v)$-square tentacle, denoted by $S_g(u,v)$. This tentacle is a small grid subgraph whose endpoints u and v (the original vertices of $G_g$) serve as connectors. An example of a $(u,v)$-square tentacle is shown in Figure 5c. Step 3: Final construction. After performing the above replacement for every such path, the resulting graph is denoted by $G_g^{1\mathrm{fd}}$ and output it.

Figure 5. Illustrations of the construction process: (a) the input grid graph $G_g$; (b) the 5-subdivision graph $G_g^{\prime }$ of $G_g$, obtained by subdividing each edge of $G_g$ into a path of length 5; (c) a square tentacle $S_g(u,v)$ used to replace the $(u,v)$-path in $G_g^{\prime }$; and (d) the resulting grid graph $G_g^{1\mathrm{fd}}$, obtained from $G_g^{\prime }$ by replacing each such path with a square tentacle. Here, double circles represent the original vertices of $G_g$, green vertices indicate vertices of the dominating set of $G_g$, and red and blue vertices denote vertices selected in the constructed 1-fair dominating set of $G_g^{1\mathrm{fd}}$ [24].

In our previous work [20,21], we introduced a placement strategy to place square tentacles on the enlarged grid graph $G_g^{\prime }$ so that any two tentacles intersect only at their connectors. The idea is summarized as follows. Consider a unit square in the grid graph $G_g$ whose vertices are $(i,j)$, $(i,j+1)$, $(i+1,j)$, and $(i+1,j+1)$. These four vertices serve as the connectors for the corresponding square tentacles. If both i and j are odd or both are even, the tentacles are placed as shown in Figure 6a; otherwise, they are arranged as in Figure 6b. When square tentacles are placed within a unit square of the enlarged grid graph $G_g^{\prime }$, at most two tentacles appear in the square, and they always occupy opposite sides. Moreover, these two tentacles are separated by at least two grid points. Consequently, no vertex other than the connectors is shared by different square tentacles. We refer to this placement scheme as Rule AT (arrangement of the tentacles).

Figure 6. The placement of square tentacles within a square of $G_g$ whose vertices are $(i,j)$, $(i,j+1)$, $(i+1,j)$, and $(i+1,j+1)$ falls into two cases: (a) both i and j are simultaneously even or odd, and (b) one of i and j is even while the other is odd.

The following lemma shows that Algorithm 1, together with Rule AT, can be executed in linear time.

Lemma 1.

Given a grid graph $G_g$, Algorithm 1 constructs a new grid graph $G_g^{1\mathrm{fd}}$ in $O(|V(G_g)|+|E(G_g)|)$-linear time.

Proof. 

We show that Algorithm 1 can be implemented in linear time and that the resulting graph is a grid graph.

Let $G_g$ be the input grid graph with $n=|V(G_g)|$ and $m=|E(G_g)|$. In Step 1, each edge of $G_g$ is subdivided four times, introducing exactly four new vertices and five edges per original edge. Consequently, the resulting graph $G_g^{\prime }$ has

$$|V(G_g^{\prime })|=n+4m\mathrm{and}|E(G_g^{\prime })|=5m,$$

and can be constructed by processing each edge once. Hence Step 1 runs in $O(n+m)$ time.

In Step 2, each subdivided path corresponding to an original edge is replaced by a constant-size tentacle $S_g(u,v)$. Since the size of each tentacle is bounded by a constant independent of n and m, each replacement requires $O(1)$ time. Applying this operation to all m edges requires $O(m)$ time.

Since Step 3 merely outputs the constructed graph $G_g^{1\mathrm{fd}}$, its running time is proportional to the output size and hence is $O(n+m)$.

Therefore, the total running time of the construction is $O(n+m)$, which is linear in the size of the input graph. Moreover, since all replacements use horizontal and vertical grid paths and grid subgraphs, the resulting graph $G_g^{1\mathrm{fd}}$ remains a grid graph.    □

We next show that the grid graph $G_g$ admits a dominating set D with $|D|\le k$ if and only if the grid graph $G_g^{1\mathrm{fd}}$ admits a 1-fair dominating set $D_{1\mathrm{fd}}$ with $|D_{1\mathrm{fd}}|\le k+2|E(G_g)|$, thereby completing the correctness of the reduction.

Before presenting the proof, we define the core of a $(u,v)$-square tentacle $S_g(u,v)$ as the set of its internal vertices, namely,

$$\mathrm{Core}(u,v):=V(S_g(u,v))\setminus \{u,v\}.$$

We first establish the only-if part as follows.

Lemma 2.

Let $G_g$ be a grid graph, and let $G_g^{1\mathrm{fd}}$ be the grid graph constructed from $G_g$ by Algorithm 1. If $G_g$ has a dominating set D with $|D|\le k$, then $G_g^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}$ with

$$|D_{1\mathrm{fd}}|\le k+2|E(G_g)|.$$

Proof. 

Let D be a dominating set of $G_g$ with $|D|\le k$. We construct a set $D_{1\mathrm{fd}}\subseteq V(G_g^{1\mathrm{fd}})$ as follows. Initially, let

$$D_{1\mathrm{fd}}:=D.$$

For each edge $(u,v)\in E(G_g)$, let $S_g(u,v)$ denote the corresponding square tentacle with core $\mathrm{Core}(u,v)$. By Rule AT, distinct tentacles intersect only at their connectors, and hence their cores are pairwise disjoint.

For each tentacle $S_g(u,v)$, select two vertices $\{w_1,w_2\}\subseteq \mathrm{Core}(u,v)$ according to the connector configuration shown in Figure 7, and update

$$D_{1\mathrm{fd}}\leftarrow D_{1\mathrm{fd}}\cup \{w_1,w_2\}.$$

Figure 7. The minimum 1-fair dominating set of square tentacle $S_g(u,v)$: (a) $u,v\in D$; (b) $u\in D$, $v\notin D$; and (c) $u,v\notin D$. Here, D is a dominating set of $G_g$ and $D_{1\mathrm{fd}}$ is a 1-fair dominating set of $G_g^{1\mathrm{fd}}$. Solid circles indicate the vertices in $D_{1\mathrm{fd}}$, and solid double circles indicate the vertices of D.

By the property of square tentacle illustrated in Figure 7, this choice ensures that every vertex of $S_g(u,v)$ not in $D_{1\mathrm{fd}}$ is adjacent to exactly one vertex of $D_{1\mathrm{fd}}$, regardless of whether the connectors u and v belong to D.

Since two vertices are added for each edge of $G_g$, we obtain

$$|D_{1\mathrm{fd}}|=|D|+2|E(G_g)|\le k+2|E(G_g)|.$$

Thus, every vertex of $G_g^{1\mathrm{fd}}$ satisfies the 1-fair domination condition, and $D_{1\mathrm{fd}}$ is a 1-fair dominating set of the desired size.

For example, Figure 5a shows a dominating set D of $G_g$ with $|D|=4$. The corresponding 1-fair dominating set $D_{1\mathrm{fd}}$ of $G_g^{1\mathrm{fd}}$, with

$$|D_{1\mathrm{fd}}|=4+2\times 11=26,$$

is illustrated in Figure 5d. □

The construction of the 1-fair dominating set $D_{1\mathrm{fd}}$ in Lemma 2 proceeds in three stages:

Stage 1: 

Process all square tentacles $S_g(u,v)$ with $u,v\in D$;

Stage 2: 

Process those with exactly one connector in D (i.e., $u\in D,v\notin D$ or $u\notin D,v\in D$);

Stage 3: 

Process the remaining square tentacles $S_g(u,v)$ with $u,v\notin D$.

By the structure of the square tentacle, we have the following lemma and observation.

Lemma 3.

Let F be a 1-fair dominating set of $G_g^{1\mathrm{fd}}$. For every edge $(u,v)\in E(G_g)$,

$$|F\cap \mathrm{Core}(u,v)|\ge 2,\mathit{where}\mathrm{Core}(u,v):=V(S_g(u,v))\setminus \{u,v\}.$$

Proof. 

The claim can be verified by inspecting the three configurations in Figure 7a–c: in each case, selecting fewer than two vertices from $\mathrm{Core}(u,v)$ leaves some internal vertex of the tentacle either undominated or adjacent to two selected neighbors, violating the 1-fair condition.    □

Observation 1.

Let F be a minimum 1-fair dominating set of $G_g^{1\mathrm{fd}}$. If $x\in V(G_g)\setminus F$ is dominated by a vertex in the tentacle $S_g(x,y)$, then $y\in F\cap V(G_g)$.

Proof. 

Figure 7b depicts this observation.    □

Next, we prove the if part in the following lemma.

Lemma 4.

Let $G_g$ be a grid graph, and let $G_g^{1\mathrm{fd}}$ be the grid graph constructed from $G_g$ by Algorithm 1. If $G_g^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}$ with $|D_{1\mathrm{fd}}|\le k+2|E(G_g)|$, then $G_g$ has a dominating set D with $|D|\le k$.

Proof. 

Let $\mathcal{F}$ be the family of all 1-fair dominating sets F of $G_g^{1\mathrm{fd}}$ such that $|F|\le k+2|E(G_g)|$. Since $G_g^{1\mathrm{fd}}$ contains a 1-fair dominating set of size at most $k+2|E(G_g)|$, we have $\mathcal{F}\ne \varnothing$. Choose $F\in \mathcal{F}$ with minimum cardinality, and define $D:=F\cap V(G_g)$.

By Lemma 3, for each $(u,v)\in E(G_g)$ we have $|F\cap \mathrm{Core}(u,v)|\ge 2$. By Rule AT, different tentacles intersect only at their connectors. Hence, the cores are pairwise disjoint. Then,

$$\left|F\cap \bigcup _{(u,v)\in E(G_g)}\mathrm{Core}(u,v)\right|\ge 2|E(G_g)|.$$

Therefore,

$$|D|=|F|-\left|F\cap \bigcup _{(u,v)\in E(G_g)}\mathrm{Core}(u,v)\right|\le |F|-2|E(G_g)|.$$

It remains to show that D dominates $G_g$. Assume to the contrary that there exists $x\in V(G_g)\setminus D$ such that $N_{G_g}(x)\cap D=\varnothing$. Since $x\notin D_{1\mathrm{fd}}$, the 1-fair condition implies that $|N_{G_g^{1\mathrm{fd}}}(x)\cap D_{1\mathrm{fd}}|=1$; let this unique neighbor be p. Because $N_{G_g}(x)\cap D=\varnothing$, we have $p\notin V(G_g)$. Thus, p lies in the tentacle $S_g(x,y)$ for some $y\in N_{G_g}(x)$. By Observation 1, $y\in F\cap V(G_g)=D$, contradicting $N_{G_g}(x)\cap D=\varnothing$. Hence D is a dominating set of $G_g$.

Finally, if $|F|\le k+2|E(G_g)|$, then $|D|\le (k+2|E(G_g)|)-2|E(G_g)|=k$. □

By Lemmas 2 and 4, we summarize the following theorem.

Theorem 2.

Let $G_g$ be a grid graph, and let $G_g^{1\mathrm{fd}}$ be the grid graph constructed from $G_g$ by Algorithm 1 and Rule AT. Then, $G_g$ has a dominating set D with $|D|\le k$ if and only if $G_g^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}$ with $|D_{1\mathrm{fd}}|\le k+2|E(G_g)|$.

The 1-fair domination problem clearly belongs to NP, since a given 1-fair dominating set can be verified in polynomial time to satisfy the 1-fair condition. By Theorem 1, Lemma 1, and Theorem 2, we obtain the following result.

Theorem 3.

The 1-fair domination problem for grid graphs is NP-complete.

By construction, the graph $G_g^{1\mathrm{fd}}$ is planar and has maximum degree 4. Thus, we obtain the following corollary.

Corollary 1.

The 1-fair domination problem is NP-complete for planar graphs with maximum degree 4.

Analogously to the construction of the grid graph $G_g^{1\mathrm{fd}}$ from $G_g$, we construct a supergrid graph $G_s^{1\mathrm{fd}}$ from $G_g$ using a different $(u,v)$-tentacle gadget as follows (see Algorithm 2).

Algorithm 2 Supergrid graph construction algorithm

Input: A grid graph $G_g$. Output: The constructed supergrid graph $G_s^{1\mathrm{fd}}$. Step 1: Enlarge $G_g$ to a grid graph $G_g^{\prime }$ by replacing each edge $(u,v)\in E(G_g)$ with a path of eight edges; that is, $G_g^{\prime }$ is the 8-subdivision of $G_g$ (see Figure 8b). Step 2: For each $(u,v)$-path in $G_g^{\prime }$, replace it with a $(u,v)$-tentacle $T_s(u,v)$, which is a small supergrid graph. The vertices u and v are called the connectors of $T_s(u,v)$ (see Figure 8c). Step 3: Let $G_s^{1\mathrm{fd}}$ be the resulting graph and output $G_s^{1\mathrm{fd}}$ (see Figure 8d).

Figure 8. (a) The grid graph $G_g$; (b) the grid graph $G_g^{\prime }$ obtained by replacing each edge of $G_g$ with a path of eight edges; (c) a $(u,v)$-tentacle $T_s(u,v)$ replacing each $(u,v)$-path in $G_g^{\prime }$; and (d) the supergrid graph $G_s^{1\mathrm{fd}}$ obtained after the replacements. Double circles represent vertices of $G_g$, and filled circles represent vertices in the dominating set of $G_g$ (respectively, the 1-fair dominating set of $G_s^{1\mathrm{fd}}$). Here, double circles represent vertices of $G_g$, green vertices indicate vertices in the dominating set of $G_g$, while red vertices denote vertices that are additionally selected in the constructed 1-fair dominating set of $G_s^{1\mathrm{fd}}$.

For example, given the grid graph shown in Figure 8a, the resulting supergrid graph $G_s^{1\mathrm{fd}}$ is shown in Figure 8d. Using arguments similar to those in the proof of Lemma 1, we obtain the following lemma.

Lemma 5.

Given a grid graph $G_g$, Algorithm 2, together with Rule AT, constructs a supergrid graph $G_s^{1\mathrm{fd}}$ in $O(|V(G_g)|+|E(G_g)|)$-linear time.

Proof. 

We show that Algorithm 2 runs in linear time and that the resulting graph is a supergrid graph.

Let $G_g$ be the input grid graph with $n=|V(G_g)|$ and $m=|E(G_g)|$. In Step 1, each edge of $G_g$ is enlarged eight times, introducing exactly seven new vertices and eight edges per original edge. Consequently, the resulting graph $G_g^{\prime }$ has

$$|V(G_g^{\prime })|=n+7m\mathrm{and}|E(G_g^{\prime })|=8m,$$

and can be constructed by processing each edge once. Hence, Step 1 runs in $O(n+m)$ time.

In Step 2, each enlarged path corresponding to an original edge is replaced by a tentacle of constant size. Since the size of each tentacle is bounded by a constant independent of n and m, each replacement requires $O(1)$ time. Applying this operation to all m edges requires $O(m)$ time.

Step 3 simply produces the resulting graph, which is linear in output size.

Therefore, the total running time of the construction is $O(n+m)$, which is linear in the size of the input graph. Moreover, according to Rule AT, each replacement in Step 2 satisfies the requirement that two distinct tentacles are separated by at least three integer lattice points (see Figure 8d). In addition, the tentacles in $G_s^{1\mathrm{fd}}$ are supergrid graphs and are pairwise disjoint, except for their connectors. Thus, the constructed graph $G_s^{1\mathrm{fd}}$ is a supergrid graph. □

The following lemma establishes a structural property of the $(u,v)$-tentacle $T_s(u,v)$ in $G_s^{1\mathrm{fd}}$.

Lemma 6.

Let F be a 1-fair dominating set of $G_s^{1\mathrm{fd}}$. For every edge $(u,v)\in E(G_g)$,

$$|F\cap (T_s(u,v)\setminus \{u,v\})|\ge 3.$$

Proof. 

This follows from the local structure of the tentacle $T_s(u,v)$ in Figure 8c: at least three internal vertices are required to satisfy the 1-fair condition for the vertices of the tentacle. □

Using the same arguments as in the proofs of Lemmas 2 and 4, we obtain the following two lemmas.

Lemma 7.

Let $G_g$ be a grid graph, and let $G_s^{1\mathrm{fd}}$ be the supergrid graph constructed from $G_g$ by Algorithm 2. If $G_g$ has a dominating set D with $|D|\le k$, then $G_s^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}^s$ with

$$|D_{1\mathrm{fd}}^s|\le k+3|E(G_g)|.$$

Proof. 

The proof follows the same construction as in Lemma 2. The differences are as follows.

• Instead of selecting two core vertices in each square tentacle, we select three internal vertices in each $(u,v)$-tentacle $T_s(u,v)$, as shown in Figure 9.
  

  Figure 9. The minimum 1-fair dominating set of tentacle $T_s(u,v)$: (a) $u,v\in D$; (b) $u\in D$, $v\notin D$; and (c) $u,v\notin D$. Here, D is a dominating set of $G_g$, and $D_{1\mathrm{fd}}^s$ is a 1-fair dominating set of $G_s^{1\mathrm{fd}}$. Solid circles represent vertices in $D_{1\mathrm{fd}}^s$, while solid double circles represent vertices of D. Green vertices indicate vertices belonging to D, and red and blue vertices denote vertices selected in $D_{1\mathrm{fd}}^s$.
• The necessity of selecting at least three internal vertices is guaranteed by Lemma 6, which states that for every $(u,v)\in E(G_g)$,

  $$|F\cap (T_s(u,v)\setminus \{u,v\})|\ge 3.$$
• Consequently, each edge contributes three vertices (instead of two), and hence

  $$|D_{1\mathrm{fd}}^s|=|D|+3|E(G_g)|\le k+3|E(G_g)|.$$

All other arguments are identical to those in Lemma 2. □

Lemma 8.

Let $G_g$ be a grid graph, and let $G_s^{1\mathrm{fd}}$ be the supergrid graph constructed from $G_g$ by Algorithm 2. If $G_s^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}^s$ with $|D_{1\mathrm{fd}}^s|\le k+3|E(G_g)|$, then $G_g$ has a dominating set D with $|D|\le k$.

Proof. 

The proof follows the same argument as in Lemma 4. The differences are as follows.

• Instead of applying Lemma 3 (which guarantees at least two core vertices in each square tentacle), we apply Lemma 6, which guarantees that each tentacle $T_s(u,v)$ contains at least three internal vertices in any 1-fair dominating set.
• Since the internal vertex sets of distinct tentacles are pairwise disjoint, we obtain

  $$\left|F\cap \bigcup _{(u,v)\in E(G_g)}(T_s(u,v)\setminus \{u,v\})\right|\ge 3|E(G_g)|.$$
• Therefore,

  $$|D|=|F|-3|E(G_g)|\le k.$$

The remaining domination argument is identical to that in Lemma 4. □

It follows from Lemmas 7 and 8 that the following theorem holds.

Theorem 4.

Let $G_g$ be a grid graph, and let $G_s^{1\mathrm{fd}}$ be the supergrid graph constructed from $G_g$ by Algorithm 2 and Rule AT. Then, $G_g$ has a dominating set D with $|D|\le k$ if and only if $G_s^{1\mathrm{fd}}$ contains a 1-fair dominating set $D_{1\mathrm{fd}}^s$ with $|D_{1\mathrm{fd}}^s|\le k+3|E(G_g)|$.

By Theorem 1, Lemma 5, and Theorem 4, we obtain the following result.

Theorem 5.

The 1-fair domination problem for supergrid graphs is NP-complete.

4. Linear-Time Algorithm for Fair Domination in Rectangular Supergrid Graphs

In this section, we present positive results for fair domination on an important restricted subclass of extended supergrid graphs, namely, rectangular supergrid graphs; while the problem is computationally intractable on general extended supergrid graphs, the strong structural regularity of the rectangular case allows efficient solutions.

4.1. Structural Properties

Let $R_{m\times n}$ denote a rectangular supergrid graph whose vertex set corresponds to an $m\times n$ rectangular region of the integer lattice. That is, $R_{m\times n}$ contains m rows and n columns of vertices. Two vertices are adjacent if they are at a unit distance horizontally, vertically, or diagonally.

Rectangular supergrid graphs exhibit a high degree of local symmetry, which allows the graph to be decomposed into small repeating patterns. In particular, the neighborhood structure of each interior vertex is identical up to translation. This uniformity plays a crucial role in the construction of fair dominating sets. Figure 10 illustrates a rectangular supergrid graph $R_{7\times 9}$.

Figure 10. A rectangular supergrid graph $R_{7\times 9}$ and its regular local structure. Horizontal edges, vertical edges, and r-skewed diagonal edges are indicated in the figure using different colors and line styles.

Lemma 9.

In a rectangular supergrid graph $R_{m\times n}$, every inner vertex has exactly eight neighbors, and the neighborhood structure is invariant under translation.

Proof. 

The claim follows directly from the definition of a rectangular supergrid graph. Each interior vertex has neighbors in the horizontal, vertical, and diagonal directions. Since the vertex lies away from the boundary, all eight such neighbors exist. Translation invariance follows from the embedding of the regular grid. □

The boundary vertices have fewer neighbors, but their neighborhoods are still highly structured. This allows boundary effects to be handled by constant-size case analysis.

4.2. A Linear-Time Algorithm for 1-Fair Domination

We now present a linear-time algorithm for computing minimum 1-fair dominating sets on rectangular supergrid graphs. The algorithm exploits the regular tiling structure of $R_{m\times n}$. We begin with the simpler case where $m=1$. Although ${\gamma }_{1\mathrm{fd}}(P_n)$ was determined in [15], we restate and prove it here for completeness, since the underlying placement structure serves as the fundamental building block for the multi-row construction developed below.

Lemma 10

([15]). ${\gamma }_{1\mathrm{fd}}(R_{1\times n})={\gamma }_{1\mathrm{fd}}(P_n)=\lceil \frac{n}{3}\rceil$.

Proof. 

Since $R_{1\times n}$ is isomorphic to the path $P_n$, it suffices to determine ${\gamma }_{1\mathrm{fd}}(P_n)$. For the path $P_n=(v_1,v_2,\dots ,v_n)$, we select one vertex from every three consecutive vertices—specifically, the middle one—to form a 1-fair dominating set. The construction is described below.

• Case 1: $n\equiv 0(\mathrm{mod}3)$ or $n\equiv 2(\mathrm{mod}3)$. If $n=3k$, let

  $$D_{1\mathrm{fd}}(P_n)=\{v_2,v_5,\dots ,v_{3i-1},\dots ,v_{3k-1}\};$$

  otherwise ($n=3k+2$), let

  $$D_{1\mathrm{fd}}(P_n)=\{v_2,v_5,\dots ,v_{3i-1},\dots ,v_{3k-1},v_{3k+2}\}.$$
• Case 2: $n\equiv 1(\mathrm{mod}3)$. In this case, let $n=3k+1$ and let

  $$D_{1\mathrm{fd}}(P_n)=\{v_1,v_3,v_6,\dots ,v_{3i},\dots ,v_{3k},v_{3k+1}\}.$$

Then, $D_{1\mathrm{fd}}(P_n)$ is a 1-fair dominating set of $P_n$ with $|D_{1\mathrm{fd}}(P_n)|=\lceil \frac{n}{3}\rceil$. Figure 11a depicts the minimum 1-fair dominating set of path $P_n$ for $1\le n\le 10$. □

Figure 11. Minimum 1-fair dominating sets of (a) $R_{1\times n}=P_n$, (b) $R_{2\times n}$, and (c) $R_{3\times n}$. Solid circles denote vertices in the minimum 1-fair dominating set.

Next, we consider $m=2$ and $m=3$. The following lemma shows that their 1-fair domination number is equal to that of $R_{1\times n}$.

Lemma 11.

${\gamma }_{1\mathrm{fd}}(R_{2\times n})={\gamma }_{1\mathrm{fd}}(R_{3\times n})=\lceil \frac{n}{3}\rceil$.

Proof. 

Consider $m=2$. Let $P_n$ be the path induced by the second row of $R_{2\times n}$, and let $D_{1\mathrm{fd}}(P_n)$ be a minimum 1-fair dominating set of $P_n$. Placing $D_{1\mathrm{fd}}(P_n)$ on the second row dominates all vertices of $R_{2\times n}$ through vertical and diagonal adjacencies, without increasing its size. Moreover, each vertex in the first row is adjacent to exactly one vertex of $D_{1\mathrm{fd}}(P_n)$. Hence, $D_{1\mathrm{fd}}(P_n)$ forms a 1-fair dominating set of $R_{2\times n}$.

In [20], we showed that $\gamma (R_{2\times n})=\lceil \frac{n}{3}\rceil$. Since every 1-fair dominating set is also a dominating set, we have $\gamma (R_{2\times n})\le {\gamma }_{1\mathrm{fd}}(R_{2\times n})$, and thus $\gamma (R_{2\times n})$ serves as a lower bound for ${\gamma }_{1\mathrm{fd}}(R_{2\times n})$. Therefore, $D_{1\mathrm{fd}}(P_n)$ placed on the second row is a minimum 1-fair dominating set of $R_{2\times n}$ of size $\lceil \frac{n}{3}\rceil$. Figure 11b illustrates such a set for $2\le n\le 10$.

By symmetry, each vertex in the third row is also adjacent to exactly one vertex of $D_{1\mathrm{fd}}(P_n)$, and the above argument extends to the case $m=3$. Figure 11c shows the minimum 1-fair dominating set of $R_{3\times n}$ for $3\le n\le 10$. This completes the proof of the lemma. □

Finally, we consider the case $n\ge m\ge 4$ and prove the following lemma.

Lemma 12.

${\gamma }_{1\mathrm{fd}}(R_{m\times n})=\lceil \frac{m}{3}\rceil \lceil \frac{n}{3}\rceil$ for $n\ge m\ge 4$.

Proof. 

Label the rows of $R_{m\times n}$ from 1 to m, where each row induces a path $P_n$. Let $D_{1\mathrm{fd}}(P_n)$ denote the selected vertex set on a row, as defined in the proof of Lemma 10. According to the value of m, we construct a 1-fair dominating set of $R_{m\times n}$ as follows.

• Case 1: $m\equiv 0(\mathrm{mod}3)$ or $m\equiv 2(\mathrm{mod}3)$. If $m=3k$, place $D_{1\mathrm{fd}}(P_n)$ on rows $2,5,\dots ,3i-1,\dots ,3k-1$. If $m=3k+2$, place $D_{1\mathrm{fd}}(P_n)$ on rows $2,5,\dots ,3i-1,\dots ,3k-1$, and $3k+2$. Figure 12a,b illustrate these placement schemes.
  

  Figure 12. Minimum 1-fair dominating sets of $R_{m\times n}$ for $n,m\ge 4$: (a) $m\equiv 0(\mathrm{mod}3)$, (b) $m\equiv 2(\mathrm{mod}3)$, and (c) $m\equiv 1(\mathrm{mod}3)$. Colored solid circles denote vertices selected in the 1-fair dominating set. The symbol ‘#’ indicates the row index of the corresponding row.
• Case 2: $m\equiv 1(\mathrm{mod}3)$. In this case, $m=3k+1$. Place $D_{1\mathrm{fd}}(P_n)$ on rows $1,3,\dots ,3i,\dots ,3k$, and $3k+1$. Figure 12c illustrates this placement.

Let $D_{1\mathrm{fd}}(R_{m\times n})$ denote the set obtained from the above construction. Then $D_{1\mathrm{fd}}(R_{m\times n})$ is a 1-fair dominating set of $R_{m\times n}$ with

$$|D_{1\mathrm{fd}}(R_{m\times n})|=\lceil \frac{m}{3}\rceil \lceil \frac{n}{3}\rceil .$$

In [20], we showed that

$$\gamma (R_{m\times n})=\lceil \frac{m}{3}\rceil \lceil \frac{n}{3}\rceil ,$$

which provides a lower bound for ${\gamma }_{1\mathrm{fd}}(R_{m\times n})$. Therefore, $D_{1\mathrm{fd}}(R_{m\times n})$ is a minimum 1-fair dominating set of $R_{m\times n}$. This completes the proof. □

It follows from Lemmas 10–12 that the following theorem holds.

Theorem 6.

For any rectangular supergrid graph $R_{m\times n}$, ${\gamma }_{1\mathrm{fd}}(R_{m\times n})=\lceil \frac{m}{3}\rceil \lceil \frac{n}{3}\rceil$ and minimum 1-fair dominating set can be computed in $O(mn)$ time.

This algorithmic construction reveals a sharp contrast between general extended supergrid graphs, where the 1-fair domination problem is NP-complete, and rectangular supergrid graphs, where strong structural regularity enables efficient solutions.

4.3. The 2-Fair Domination in Rectangular Supergrid Graphs

In this subsection, we compute ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$ for the rectangular supergrid graph $R_{m\times n}$ with $m=1$ and 2. We first compute ${\gamma }_{2\mathrm{fd}}(R_{1\times n})$ as follows.

Lemma 13.

For any rectangular supergrid graph $R_{1\times n}$, ${\gamma }_{2\mathrm{fd}}(R_{1\times n})=\lfloor \frac{n}{2}\rfloor +1$.

Proof. 

Observe that $R_{1\times n}$ is the path graph $P_n=(v_1,v_2,\dots ,v_n)$. Let D be a 2-fair dominating set of $P_n$. Since the endpoints $v_1$ and $v_n$ have a degree 1, they must belong to D; otherwise, they would be adjacent to at most one vertex of D, violating the 2-fair condition. Hence, $v_1,v_n\in D$.

For any internal vertex $v_i$ with $2\le i\le n-1$, if $v_i\notin D$, then it must be adjacent to exactly two vertices in D, which forces $v_{i-1},v_{i+1}\in D$. Consequently, no two vertices outside D can be adjacent. Thus, $V(P_n)\setminus D$ is an independent set contained in $\{v_2,\dots ,v_{n-1}\}$, whose maximum size is $\lceil \frac{n-2}{2}\rceil$. Therefore,

$$|D|\ge n-\lceil \frac{n-2}{2}\rceil =\lfloor \frac{n}{2}\rfloor +1.$$

On the other hand, let

$$D_{2\mathrm{fd}}=\{v_1,v_3,v_5,\dots \}\cup \left\{v_n\right\}.$$

Every vertex not in $D_{2\mathrm{fd}}$ is adjacent to exactly two vertices of $D_{2\mathrm{fd}}$, and hence $D_{2\mathrm{fd}}$ is a 2-fair dominating set. Its cardinality is $\lfloor \frac{n}{2}\rfloor +1$, establishing the equality. Figure 13a depicts the constructed minimum 2-fair dominating set of ${\gamma }_{2\mathrm{fd}}(R_{1\times n})$ for $1\le n\le 10$. □

Figure 13. Minimum 2-fair dominating sets of (a) $R_{1\times n}$ and (b) $R_{2\times n}$. Solid circles indicate vertices in the 2-fair dominating sets. Solid circles indicate vertices in the 2-fair dominating sets. The dashed blue lines indicate the starting positions of the repeating construction pattern, and the dashed yellow box highlights the selected vertices on the first row in the construction for $R_{2\times n}$.

Next, consider $m=2$. We compute ${\gamma }_{2\mathrm{fd}}(R_{2\times n})$ as the following lemma.

Lemma 14.

For any rectangular supergrid graph $R_{2\times n}$ with $2\le n$, ${\gamma }_{2\mathrm{fd}}(R_{2\times n})=2\lceil \frac{n}{3}\rceil$.

Proof. 

Denote by $a_j$ and $b_j$ the vertices in the upper and lower positions of column j, respectively. In $R_{2\times n}$, every pair of consecutive columns induces a complete graph $K_4$.

Let D be a 2-fair dominating set of $R_{2\times n}$. Consider any three consecutive columns $j,j+1,j+2$. If column $j+1$ contains no vertex of D, then each of its two vertices must be adjacent to exactly two vertices of D, which is possible only if at least two vertices of D appear in columns j and $j+2$. If column $j+1$ contains exactly one vertex of D, then the other vertex in column $j+1$ has only one neighbor in D within column $j+1$ and therefore must have at least one additional neighbor in D from column j or column $j+2$. Consequently, in either case, every three consecutive columns contain at least two vertices of D.

Hence,

$$|D|\ge 2\lceil \frac{n}{3}\rceil ,$$

so $2\lceil \frac{n}{3}\rceil$ is a lower bound for ${\gamma }_{2\mathrm{fd}}(R_{2\times n})$.

To show the upper bound, we construct a 2-fair dominating set $D_{2\mathrm{fd}}$ as follows. Let $n=3k$, $3k+1$, or $3k+2$ with $k\ge 1$. Select both vertices in columns $2,5,8,\dots ,3k-1$, i.e.,

$$D_{2\mathrm{fd}}=\bigcup _{t=1}^k\{a_{3t-1},b_{3t-1}\}.$$

If $n=3k+1$, additionally select both vertices in column $3k$; if $n=3k+2$, additionally select both vertices in column n (see Figure 13b). It is straightforward to verify that every vertex not in $D_{2\mathrm{fd}}$ has exactly two neighbors in $D_{2\mathrm{fd}}$. Thus, $D_{2\mathrm{fd}}$ is a 2-fair dominating set of size $2\lceil \frac{n}{3}\rceil$, completing the proof. □

Remark 1.

Although any two consecutive columns induce a $K_4$, the 2-fair condition does not imply that every two consecutive columns must contain two vertices of D: two selected vertices may be concentrated in a single column within a three-column block (see Figure 13b).

Lemmas 13 and 14 determine the exact values of the 2-fair domination number of $R_{m\times n}$ for the special cases $m=1$ and $m=2$. These results illustrate the tractable boundary cases of the problem and provide insight into the structural behavior of 2-fair domination on rectangular supergrid graphs. For larger values $m\ge 3$, it was claimed in [22] that

$${\gamma }_{2\mathrm{fd}}(R_{m\times n})=mn,$$

that is, the entire vertex set $V(R_{m\times n})$ is the unique 2-fair dominating set. However, using an ILP model to compute minimum 2-fair dominating sets for small instances with $n\ge m\ge 3$, we find that this claim does not hold in general. For example, ${\gamma }_{2\mathrm{fd}}(R_{5\times 5})=8$ and ${\gamma }_{2\mathrm{fd}}(R_{8\times 8})=18$, as illustrated in Section 5.4. Nevertheless, we are unable to compute ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$ efficiently for larger instances (e.g., when $n\ge m\ge 3$). Determining the computational complexity and deriving a general formula for ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$ with $n\ge m\ge 3$ remain open problems.

5. ILP Formulation for the k-Fair Domination Problem

In this section, we formulate the k-fair domination problem as an integer linear programming (ILP), which is used as a verification tool for small instances. The proposed ILP formulation is a 0–1 integer linear programming, since each variable represents whether a vertex is selected in the fair dominating set. This ILP model applies not only to extended supergrid graphs but also to arbitrary graphs. Due to its inherent computational complexity, the ILP formulation is employed only for small instances.

For the k-fair domination problem on G, we give the following remark.

Remark 2.

If $k>\Delta$, then any vertex $v\notin D_{k\mathrm{fd}}$ would require $|N_G(v)\cap D_{k\mathrm{fd}}|=k>de{g}_G(v)$, which is impossible. Hence, the only feasible k-fair dominating set is $D_{k\mathrm{fd}}=V(G)$.

Therefore, we assume $1\le k\le \Delta$, since case $k>\Delta$ admits only the trivial solution $D_{k\mathrm{fd}}=V(G)$. In this paper, we use the ILP formulation with $k=1$ or $k=2$; that is, we focus on the 1-fair and 2-fair domination problems.

5.1. ILP Formulation

Let G be a graph with maximum degree $\Delta$. For each vertex $v\in V(G)$, we introduce a binary decision variable

$$x_v=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}v\mathrm{is}\mathrm{selected}\mathrm{in}\mathrm{the}k-\mathrm{fair}\mathrm{dominating}\mathrm{set}{D}_{k\mathrm{fd}};\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Therefore, the introduced model is a 0-1 integer linear programming (ILP) formulation.

5.1.1. General ILP Formulation

The k-fair domination problem on graph G can be formulated as the following integer linear programming.

• Objective function:

  $$\min \sum _{v\in V(G)}{x}_v.$$
• Constraints:

  $$\begin{array}{cc}\sum _{u\in N_G\left[v\right]}{x}_u\ge 1,\hfill & \hfill \forall v\in V(G),\end{array}$$

  (1)

  $$\begin{array}{cc}\sum _{u\in N_G(v)}{x}_u\ge k(1-x_v),\hfill & \hfill \forall v\in V(G),\end{array}$$

  (2)

  $$\begin{array}{cc}\sum _{u\in N_G(v)}{x}_u\le k+(\Delta -k)x_v,\hfill & \hfill \forall v\in V(G),\end{array}$$

  (3)

  $$\begin{array}{cc}{x}_v\in \{0,1\},\hfill & \hfill \forall v\in V(G).\end{array}$$

  (4)
• Correctness:

Constraint (1) ensures that every vertex is dominated (domination condition). Constraints (2) and (3) (the k-fair condition) together enforce that each vertex $v\notin D_{k\mathrm{fd}}$ is adjacent to exactly k vertices of $D_{k\mathrm{fd}}$, while no restriction is imposed when $v\in D_{k\mathrm{fd}}$. Therefore, every feasible solution corresponds to a k-fair dominating set of G, and conversely. Note that $D_{k\mathrm{fd}}=\{v\in V(G)\mid x_v=1\}$.

5.1.2. Refined ILP Formulation (Compact but Strengthened Model)

The general ILP formulation can be made more compact by incorporating the fairness constraint directly into a modified adjacency matrix. Let $A^{\prime }=(a_{vu}^{\prime })$ be defined by

$$a_{vv}^{\prime }=k\mathrm{for}\mathrm{all}v\in V(G),a_{vu}^{\prime }=1\mathrm{if}u\in N_G(v),a_{vu}^{\prime }=0\mathrm{otherwise}.$$

Then Constraints (1)–(3) can be replaced by the single equality

$$\begin{array}{c}\hfill \sum _{u\in V(G)}{a}_{vu}^{\prime }{x}_u=k,\forall v\in V(G).\end{array}$$

(5)

This compact formulation reduces the number of constraints and yields a more memory-efficient model. However, it is important to note that Constraint (5) is strictly stronger than the original k-fair domination formulation.

Indeed, if $x_v=1$, then Constraint (5) implies

$$k+\sum _{u\in N_G(v)}{x}_u=k,$$

and hence ${\sum }_{u\in N_G(v)}{x}_u=0$. Therefore, any two selected vertices must be nonadjacent, which means that the selected set forms an independent set. This additional restriction is not required in the definition of k-fair domination and may render the refined model infeasible even when a valid k-fair dominating set exists.

For this reason, the general ILP formulation is retained as the primary model to guarantee correctness and feasibility for arbitrary instances. The refined formulation may be used as a strengthened (optional) variant in cases where the independent-set property is known to hold. In our computational experiments, the refined model was observed to be faster on instances where feasibility is preserved, but the general model remains necessary to ensure robustness.

5.1.3. Implementation Note

We first attempt to solve the fair domination problem by using the refined ILP model (Constraint (5)). If it is infeasible, we fall back to the general ILP formulation, which is always feasible since $V(G)$ is a trivial k-fair dominating set.

5.2. ILP-Based Experiments

We implement the general ILP model to compute minimum 1-fair and 2-fair dominating sets on small supergrid graphs.

5.2.1. Holes (Vertex Deletions)

Let $R_{m\times n}$ be an $m\times n$ supergrid graph with vertex set

$$V(R_{m\times n})=\{(i,j)\mid 1\le i\le n,1\le j\le m\}.$$

$$R_{m\times n}^p=R_{m\times n}[V(R_{m\times n})\setminus H],$$

which is a supergrid graph obtained from $R_{m\times n}$ with hole rate p.

5.2.2. Experimental Setup

In our experiments, the set H is generated uniformly at random among all subsets of size $\lfloor p·mn\rfloor$, and a fixed random seed is used to ensure reproducibility. We also guarantee that at least one vertex remains after deletions, that is, $|V(R_{m\times n})\setminus H|\ge 1$, and that the resulting supergrid graph $R_{m\times n}^p$ is connected.

All test instances were generated from $m\times n$ rectangular supergrid graphs by applying the above vertex-deletion process. The ILP formulation was solved to optimality using the CBC solver through the PuLP interface. Due to its computational complexity, the general ILP model was employed only for small instances and served as an exact experimental tool for investigating the structural behavior of 1-fair and 2-fair domination under vertex deletions.

5.3. ILP Formulation for 1-Fair Domination on Supergrid Graphs

While the theoretical results in the previous sections provide structural characterizations and exact values for several classes of supergrid graphs, the behavior of the 1-fair domination number under structural perturbations remains unclear. In particular, the strict local balance condition of 1-fair domination suggests that even small changes in the graph structure may significantly affect the size and configuration of feasible solutions.

To gain further insight into this phenomenon, we adopt an integer linear programming (ILP) approach to compute minimum 1-fair dominating sets on small supergrid graphs. The ILP formulation allows us to systematically explore how the domination number changes when the graph is subjected to random vertex removals, and to observe structural patterns that are difficult to capture by purely theoretical analysis.

5.3.1. Experimental Setup

In our experiments, we consider the rectangular supergrid graph $10\times 10$, which contains horizontal, vertical, and diagonal edges. To model structural defects, we introduce holes by randomly removing a prescribed proportion of vertices together with their incident edges. The hole rate p is chosen from the set

$$p\in \{0.00,0.15,0.20,0.25,0.30,0.35\}.$$

For each hole rate, a general ILP model is used to compute a minimum 1-fair dominating set $D_{1\mathrm{fd}}$, where every vertex not in $D_{1\mathrm{fd}}$ is adjacent to exactly one vertex of $D_{1\mathrm{fd}}$. The objective function minimizes $|D_{1\mathrm{fd}}|$. Representative optimal solutions for different hole rates are illustrated in Figure 14, where the selected vertices are marked in solid circles.

Figure 14. The minimum 1-fair dominating set of supergrid graph constructed from $R_{10\times 10}$ by removing p hole rate of vertices for $p\in \{0.00,0.15,0.20,0.25,0.30,0.35\}$, where solid circles indicate the vertices in the 1-fair dominating set.

5.3.2. Analysis of Results

The experimental results reveal a pronounced non-monotonic behavior of the minimum 1-fair dominating set size with respect to the hole rate. When no vertices are removed ($p=0$), the $10\times 10$ supergrid graph admits a compact 1-fair dominating set $D_{1\mathrm{fd}}$ of size 16. However, for moderate hole rates ($p=0.15$, $0.20$, and $0.25$), the size of a minimum 1-fair dominating set increases dramatically, reaching values as large as 85 (the entire vertex set of $10\times 10$ supergrid graph with hole rate $0.15$).

This phenomenon can be explained by the strict local balance imposed by the 1-fair constraint. For moderately perturbed graphs, although global connectivity is largely preserved, vertex removals create irregular local neighborhoods. These irregularities often force the ILP solver to include additional dominating vertices in order to avoid multiple domination or violations of the fairness constraint, thus substantially increasing $|D_{1\mathrm{fd}}|$.

Interestingly, when the hole rate becomes sufficiently large ($p=0.30$), the size of the minimum 1-fair dominating set decreases again to 15. In this case, the graph typically decomposes into smaller or sparsely connected components with reduced vertex degrees, which are easier to satisfy the 1-fair condition. When the hole rate increases further to $p=0.35$, the domination size increases again to 62, reflecting another structural transition of the graph.

Overall, these results indicate that the difficulty of the 1-fair domination problem on supergrid graphs is highly sensitive to intermediate levels of structural disruption. Moderate removals of the vertex can significantly increase the domination cost, whereas more extensive removals can simplify the structure of the graph and reduce the size of a 1-fair minimum dominating set. We summarize these findings as follows.

Observation 2.

For supergrid graphs with vertex holes, the size of a minimum 1-fair dominating set is not monotone with respect to the hole rate. In particular, moderate hole rates may substantially increase the domination cost due to local violations of the fairness constraint, while higher hole rates may simplify the graph structure and lead to smaller 1-fair dominating sets.

Observation 3.

For supergrid graphs with vertex deletions, an optimal solution of the 1-fair domination problem may approach selecting almost all remaining vertices. Figure 14 shows that nearly the entire vertex set is selected into the 1-fair dominating set when $p=0.15,0.20,0.25,$ and $0.35$. This phenomenon is caused by low-degree vertices whose local neighborhoods are severely disrupted by holes.

The above observations indicate that the degenerate behavior is not an artifact of the ILP solver, but an intrinsic consequence of imposing an exact local fairness constraint on irregular graph structures. Specifically, the requirement that every vertex outside the dominating set must be adjacent to exactly one dominating vertex forces the ILP to include many vertices in the dominating set in order to avoid violations caused by disrupted local neighborhoods.

To address this issue, we introduce a restricted version of k-fair domination that excludes vertices with insufficient local degree from the fairness constraint. This modification prevents the ILP from selecting vertices solely to satisfy local fairness constraints, while preserving the exact fairness requirement on structurally stable vertices.

5.3.3. Restricted k-Fair Domination

Definition 1.

Let G be a graph, let $k\ge 1$ be an integer, and let $\tau \ge k+1$ be a fixed integer. For each vertex $v\in V(G)$, define

$${\alpha }_v=\left\{\begin{array}{cc}1,\hfill & \mathit{if}de{g}_G(v)\ge \tau ,\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

A set $D\subseteq V(G)$ is called a $\tau$-restricted k-fair dominating set of G if the following conditions hold:

1. 

Domination. For every vertex $v\in V(G)$,

$$N_G\left[v\right]\cap D\ne ⌀.$$

2. 

Restricted fairness. For every vertex $v\in V(G)\setminus D$ with $de{g}_G(v)\ge \tau$,

$$|N_G(v)\cap D|=k.$$

• No fairness requirement is imposed on vertices $v\in V(G)\setminus D$ with $de{g}_G(v)<\tau$.

Remark 3.

When $\tau =0$, we have ${\alpha }_v=1$ for all $v\in V(G)$, and the restricted k-fair domination reduces to the classical k-fair domination problem.

The above definition admits a formulation of natural integer linear programming by activating the fairness constraint only on vertices with a degree of at least $\tau$. Specifically, the ILP formulation for the restricted problem can be obtained by replacing Constraints (2) and (3) with the following constraints:

$$\begin{array}{cc}\sum _{u\in N_G(v)}{x}_u\ge k{\alpha }_v(1-x_v),\hfill & \hfill \forall v\in V(G),\end{array}$$

(6)

$$\begin{array}{cc}\sum _{u\in N_G(v)}{x}_u\le k{\alpha }_v(1-x_v)+d(v)x_v+d(v)(1-{\alpha }_v),\hfill & \hfill \forall v\in V(G),\end{array}$$

(7)

where $d(v)=de{g}_G(v)$.

The ILP formulation is modified by activating the fairness constraint only on vertices with a degree of at least $\tau$, in accordance with Definition 1.

5.3.4. Motivation and Feasibility of Restricted Fair Domination

The classical k-fair domination imposes an exact local balance condition: every vertex outside the dominating set must have exactly k neighbors in it. Under vertex deletions, this equality constraint becomes extremely sensitive to irregular local neighborhoods, and optimal solutions may be forced to include a large fraction of vertices merely to avoid fairness violations. To mitigate this degeneracy, we introduce $\tau$-restricted k-fair domination, which enforces the fairness equality only on vertices with degree at least $\tau$, while keeping the domination requirement for all vertices.

This restriction enlarges the feasible region: any k-fair dominating set (if it exists) is also a $\tau$-restricted k-fair dominating set for any $\tau$. Moreover, if $\tau >\Delta (G)$, then ${\alpha }_v=0$ for all $v\in V(G)$ and the restricted problem reduces to the domination problem; hence a feasible solution exists whenever G is nonempty.

Proposition 1

(a sufficient condition for degeneracy). Let G be a graph, and let $k\ge 1$. Suppose that a vertex $v\in V(G)$ satisfies $de{g}_G(v)<k$. Then, in any k-fair dominating set D of G, we must have $v\in D$. Consequently, if $L=\{v\in V(G)\mid de{g}_G(v)<k\}$, then every k-fair dominating set D satisfies $L\subseteq D$, and hence $|D|\ge |L|$.

Proof. 

Assume for the contradiction that $v\notin D$. Since D is k-fair, v must be adjacent to exactly k vertices of D, that is, $|N_G(v)\cap D|=k$. However, $|N_G(v)\cap D|\le |N_G(v)|=de{g}_G(v)<k$, a contradiction. Therefore, $v\in D$. The second statement follows immediately. □

Proposition 1 shows that low-degree vertices ($de{g}_G(v)<k$) are forced to be selected in any feasible solution of the classical k-fair domination problem. Under vertex deletions, many vertices near holes may become low-degree, which explains why the optimal solution can contain a large fraction of vertices. This motivates our $\tau$-restricted model, where such structurally unstable vertices are excluded from the fairness requirement.

Corollary 2.

If $\tau \ge k+1$, then vertices with $de{g}_G(v)<\tau$ are not subject to the fairness constraint in Definition 1, and thus are no longer forced to be included solely due to the infeasibility of $|N_G(v)\cap D|=k$.

5.3.5. Experimental Results for the Restricted 1-Fair Domination

Figure 15 reports the ILP solutions for the $\tau$-restricted 1-fair domination problem on the $10\times 10$ supergrid graph (using the same instances as in Figure 14), where $\tau =5$ and different hole rates are considered. When no vertices are removed, the restricted model produces a minimum dominating set of size $|D_{\mathrm{r}1\mathrm{fd}}|=16$, which coincides with the optimal solution of the original 1-fair domination problem. This indicates that restricting the scope of fairness enforcement does not affect the optimal solution in regular graph structures.

Figure 15. The minimum restricted 1-fair dominating set of the supergrid graph constructed from $R_{10\times 10}$ by removing vertices with hole rates $p\in \{0.00,0.15,0.20,0.25,0.30,0.35\}$, where solid circles indicate the vertices in the restricted 1-fair dominating set.

For moderate hole rates, the restricted formulation exhibits a more stable behavior than the original 1-fair model. In particular, for hole rates $p=0.15$ and $0.20$, the restricted model still produces solutions of sizes 66 and 59, respectively, which are smaller than those of the original 1-fair model. When the hole rate increases to $p=0.25$, the size of the restricted solution drops significantly to 19, while the original 1-fair model still yields a much larger solution.

When the hole rate increases further ($p=0.30$ and $p=0.35$), the restricted model continues to produce compact solutions of sizes 15 and 23, respectively, both of which remain close to the baseline value. This behavior contrasts with the original 1-fair domination problem, where the solution size may remain very large even at higher hole rates.

Overall, these results demonstrate that restricting the enforcement of fairness constraints effectively mitigates the extreme growth and instability observed in the original 1-fair domination problem, leading to solutions that are more robust and predictable under structural perturbations of supergrid graphs.

5.4. ILP Formulation for 2-Fair Domination on Supergrid Graphs

In Lemmas 13 and 14, we compute ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$ for the rectangular supergrid graph $R_{m\times n}$ with $m=1$ and 2. For $m\ge 3$, we claimed in [22] that ${\gamma }_{2\mathrm{fd}}(R_{m\times n})=mn$, that is, the entire vertex set $V(R_{m\times n})$ is the unique 2-fair dominating set. In this subsection, we will use an ILP model to give counterexamples for it.

Figure 16 shows the minimum 2-fair dominating sets of $R_{n\times n}$ with $5\le n\le 9$ and $n=11$ via general ILP solver. On the other hand, Figure 16 also shows that many rectangular supergrid graphs admit no proper 2-fair dominating set; that is, their unique 2-fair dominating set is the entire vertex set. This phenomenon is analogous to that observed in the 1-fair domination problem. However, ${\gamma }_{2\mathrm{fd}}(R_{n\times n})\ne n\times n$ for $n\in \{5,8,11\}$ in Figure 16 and hence the claim in [22] that ${\gamma }_{2\mathrm{fd}}(R_{m\times n})=mn$ is not true. Currently, we are unable to identify a general structural rule computing ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$. Therefore, determining the value of ${\gamma }_{2\mathrm{fd}}(R_{m\times n})$ for $n\ge m\ge 3$ remains an open problem.

Figure 16. The minimum 2-fair dominating set of $R_{n\times n}$ for $5\le n\le 9$ and $n=11$, where solid circles indicate the vertices in the 2-fair dominating set.

Finally, we use general ILP solver to compute minimum 2-fair dominating sets for small supergrid graphs. The experimental setting is identical to that used for the 1-fair domination problem. Figure 17 shows the minimum 2-fair dominating sets for the supergrid graphs listed in Figure 14.

Figure 17. The minimum 2-fair dominating set of supergrid graph constructed from $R_{10\times 10}$ by removing p hole rate of vertices for $p\in \{0.00,0.15,0.20,0.25,0.30,0.35\}$, where solid circles indicate the vertices in the 2-fair dominating set.

Similar to the 1-fair domination case, there exist graphs G for which the 2-fair constraint is so restrictive that the entire vertex set $V(G)$ becomes the only feasible 2-fair dominating set. This happens because each vertex not in the dominating set must be adjacent to exactly two vertices in the set. In some graphs, especially those with boundary vertices or irregular local degrees (such as graphs with holes), this requirement cannot be satisfied unless every vertex is selected. Consequently, the trivial solution $D=V(G)$ becomes the unique feasible 2-fair dominating set. This phenomenon is also observed in the ILP results. For example, the $10\times 10$ rectangular supergrid without holes yields $|D_{2\mathrm{fd}}|=100$, meaning that all vertices must be selected.

6. Conclusions and Open Problems

In this paper, we studied fair domination on extended supergrid graphs. We established that the 1-fair domination problem is NP-complete for grid graphs, supergrid graphs, and planar graphs with maximum degree 4. These results demonstrate that imposing an exact local fairness constraint does not simplify the domination problem on grid-like graph classes.

On the positive side, we showed that restricting attention to rectangular supergrid graphs yields a markedly different landscape. Taking advantage of their strong regularity, we present a linear-time algorithm for computing minimum 1-fair dominating sets on rectangular supergrid graphs. We also solve the 2-fair domination problem on rectangular supergrid graphs with one and two rows of vertices in linear time.

In addition, we formulate a unified ILP model for the k-fair domination problem and use it as an exact experimental tool on small supergrid instances with vertex deletions (holes). The results indicate that the k-fair constraint can be so restrictive that the trivial solution $D=V(G)$ becomes the only feasible k-fair dominating set in certain perturbed instances. Motivated by this degenerate behavior, we introduce the notion of restricted k-fair domination, which enforces fairness only on vertices of sufficiently large degree, thereby improving robustness under structural irregularities.

Several interesting questions remain open. In particular, while domination is known to be NP-complete on grid graphs, the computational complexity of the 2-fair (and more generally, k-fair for $k\ge 2$) domination problem on grid graphs and extended supergrid graphs is still unknown. Moreover, the complexity of the 2-fair domination problem on $R_{m\times n}$ with $n\ge m\ge 3$ is still unknown. Resolving these questions would further clarify the role of structural constraints in the complexity of fair domination. The computational complexity and structural properties of the restricted k-fair domination problem remain open. In particular, determining whether restricted fair domination is polynomial-time solvable or NP-complete on grid-like graph classes is an interesting direction for future research.