Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

: For a simple graph G = ( V , E ) with no isolated vertices, a total Roman {3}-dominating func-tion(TR3DF) on G is a function f : V ( G ) → { 0,1,2, 3 } having the property that (i) ∑ w ∈ N ( v ) f ( w ) ≥ 3 if f ( v ) = 0; (ii) ∑ w ∈ N ( v ) f ( w ) ≥ 2 if f ( v ) = 1; and (iii) every vertex v with f ( v ) (cid:54) = 0 has a neighbor u with f ( u ) (cid:54) = 0 for every vertex v ∈ V ( G ) . The weight of a TR3DF f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γ t { R 3 } ( G ) . In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γ t { R 3 } for trees.


Introduction
Let G = (V, E) be a graph with vertex set V = V(G) and edge set E = E(G). A vertex of degree one is called a leaf and its neighbor is a support vertex, and a support vertex is called a strong support if it is adjacent to at least two leaves. Let S n be a star with order n. A tree T is an acyclic connected graph. G = (G 1 ∪ G 2 ) is a union graph G such that V(G) = V(G 1 ) ∪ V(G 2 ) and E(G) = E(G 1 ) ∪ E(G 2 ).
A subset S of a vertex set V(G) is a dominating set of G if for every vertex v ∈ V(G) \ S, there exists a vertex w ∈ S such that wv is an edge of G. The domination number of G denoted by γ(G) is the smallest cardinality of a dominating set S of G [1]. A function [2]. The dominating set problem(DSP) is to find the domination number of G, which has been deeply and widely studied in recent years [3][4][5][6][7].
A subset S of a vertex set V(G) is a total dominating set of G if v∈S N(v) = V(G). The total domination number of G denoted by γ t (G) is the smallest cardinality of a total dominating set S of G [8]. The literature on the subject of total domination in graphs has been surveyed and provided in detail in a recent book [9]. Moreover, Michael A. Henning et al. presented a survey of selected recent results on total domination in graphs [10].
The mathematical concept of Roman domination is originally defined and discussed by Stewart et al. [11] and ReVelle et al. [12]. A Roman dominating function(RDF) on graph G is a function f : V(G) → {0, 1, 2} such that every vertex v ∈ V(G) for which f (u) = 0 is adjacent to at least one vertex u with f (u) = 2 [13]. The Roman domination number of G is the minimum weight overall RDFs, denoted by γ R (G) [14]. On the basis of Roman domination, signed Roman domination [15], double Roman domination [16] and total Roman domination [17] have been proposed recently.
The total Roman dominating function(TRDF) on G is an RDF f on G with an additional property that every vertex v ∈ V(G) with f (v) = 0 has a neighbor u with f (u) = 0. Let γ tR (G) denote the minimum weight of all TRDFs on G. A TRDF on G with weight γ tR (G) is called a γ tR (G)-function. The conception of TRDF was first defined by Hossein Ahangar et al. [18]. In addition, Nicolás Campanelli et al. studied the total Roman domination number of the lexicographic product of graphs [17] and Chloe Lampman et al. presented some basic results of Edge-Critical Graphs [19].
The Roman {2}-dominating function (also named Italian domination) f [20] introduced by Chellali et al. which is defined as follows: Chellali et al. proved that the Roman {2}-domination problem is NP-complete for bipartite graphs [21]. Hangdi Chen showed that the Roman {2}-domination problem is NP-complete for split graphs, and gave a linear-time algorithm for finding the minimum weight of Roman {2}-dominating function in block graphs [22]. As a generalization of Roman domination, Michael A. Henning et al. studied the relationship between Roman {2}-domination and dominating set parameters in trees [20].
A Roman {3}-dominating function(R{3}DF) f defined by Mojdeh et al. [23], which is defined as follows: presented an upper bound on the Roman {3}-domination number of a connected graph G, characterized the graphs attaining upper bound and showed that the Roman {3}-domination problem is NP-complete, even restricted to bipartite graphs [23] .
The total Roman {3}-domination [24] was studied recently . The total Roman {3}-dominating function(TR3DF) on a graph G is an R{3}DF on G with the additional property that every vertex v ∈ V(G) with f (v) = 0 has a neighbor w with f (w) = 0. The minimum weight of a total Roman {3}-dominating function on G denoted by γ t{R3} (G) is named the total Roman {3}-domination number of G. A γ t{R3} (G)-function is a total Roman {3}-dominating function on G with weight γ t{R3} (G). Doost Ali Mojdeh et al. showed the relationship among total Roman {3}-domination, total domination, and total Roman{2}-domination parameters. They also presented an upper bound on the total Roman {3}-domination number of a connected graph G and characterized the graphs arriving this bound. Finally, they investigated that total Roman {3}-domination problem is NP-complete for bipartite graphs [24].
In this paper, we further investigate the complexity of total Roman {3}-domination in planar graphs and chordal bipartite graphs. Moreover, we give a linear-time algorithm to compute the γ t{R3} for trees which answer the problem that it is possible to construct a polynomial algorithm for computing the number of total Roman {3}-domination for trees [24].

Complexity
In this section, we study the complexity of total Roman {3}-domination of graph. We show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Consider the following decision problem. Please note that the dominating set problem is NP-complete for planar graphs [25] and chordal bipartite graphs [26]. We show the NP-completeness results by reducing the well-known NP-complete problem, dominating set, to TR3D.
Let G be a graph on n vertices. Let T v be the tree with V( Figure 1.

Lemma 1.
If G is a planar graph or chordal bipartite graph , so is G .
Lemma 4. If f is a DF of G with ω f (G) ≤ , then there exists a TR3DF g of G with ω g (G ) ≤ + 8n.

Proof of Lemma 4.
For each v ∈ V(G), we define g as follows: of G . Therefore we have that ω g (G ) = ω f (G) + 8n ≤ + 8n. Claim 1. Let g be a TR3DF of G , then ω g (T v ) ≥ 8.

Proof of Claim 2. By the definition of
If h(v) = 0, then we define g : Figure 2. Therefore g is a TR3DF of G such that g(v a ) + g(v b ) ≤ 2 and ω g (G ) = ω h (G ).
If h(v) ≥ 1, then we define g :

Lemma 5.
If g is a TR3DF of G with ω g (G ) ≤ + 8n, then there exists a DF f of G with ω f (G) ≤ .

A Linear-Time Algorithm for Total Roman {3}-Domination in Trees
In this section, we present a linear-time algorithm to compute the minimum weight of total Roman {3}-dominating function for trees. First, we define the following concepts:

Definition 2. The minimum weight overall F
u,G number of G, and a γ

Lemma 6.
For any graph G with specific vertex u, we have Lemma 7. Suppose T 1 and T 2 are trees with specific vertices v and u, respectively. Let T 3 be the tree with the specific vertex u, which is obtained by joining a new edge uv from the union of T 1 and T 2 , as depicted in Figure 3. Then the following statements hold for γ (a) For i = 0, j ∈ {0, 1, 2, 3}, we have : (v, T 1 )-function of T 1 ,for s ∈ {0, 1, 2}.
Lemmas 6 and 7 give the following dynamic programming algorithm 1 for the total Roman {3}-domination problem in trees.

Conclusions
The total Roman {3}-domination problem was introduced and studied in [24] , and it was proven to be NP-complete for bipartite graphs. In this paper , we prove that the total Roman {3}-domination problem is NP-complete for planar graphs or chordal bipartite graphs , and showed a linear-time algorithm for total Roman {3}-domination problem on trees. For the algorithmic aspects of the total Roman {3}-domination problem , designing exact algorithms or approximation algorithms on general graphs , or polynomial algorithms for total Roman {3}-domination problem on some special classes graphs deserve further research.
Author Contributions: Conceptualization, X.L., H.J. and Z.S.; writing, X.L. and Z.S.; review, H.J. and Z.S.; investigation: P.W. All authors have contributed equally to this work. All authors have read and agreed to the possible publication of the manuscript.
Funding: This work is supported by the Natural Science Foundation of Guangdong Province under Grant 2018A0303130115.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: