Characterizations of Minimal Dominating Sets in γ -Endowed and Symmetric γ -Endowed Graphs with Applications to Structure-Property Modeling

: Claude Berge (1987) introduced the concept of k -extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph H = ( V , E ) . Graphs possessing the property of being 1-extendable are termedas Berge graphs. This introduction gave rise to the notion of well-covered graphs and well-dominated graphs. A graph is categorized as well-covered if each of its maximal independent sets is, in fact, a maximum independent set. Similarly, a graph attains the classification of well-dominated if every minimal dominating set (DS) within it is a minimum dominating set. In alignment with the concept of k -extendable graphs, the framework of ( k , γ ) -endowed graphs and symmetric ( k , γ ) -endowed graphs are established. In these graphs, each DS of size k encompasses a minimum DS of the graph. In this article, a study of γ -endowed dominating sets is initiated. Various results providing a deep insight into γ -endowed dominating sets in graphs such as those characterizing the ones possessing a unique minimum DS are proven. We also introduce and study the symmetric γ -endowed graphs and minimality of dominating sets in them. In addition, we give a solution to an open problem in the literature. which seeks to find a domination-based parameter that has a correlation coefficient of ρ > 0.9967 with the total π -electronic energy of lower benzenoid hydrocarbons. We show that the upper dominating number Γ ( H ) studied in this paper delivers a strong prediction potential.


Introduction
All the undefined notations and terminologies have been introduced in Section 2. Domination-related graphical parameters have immense applications in computer science, engineering and chemistry.The earliest domination-related parameters were reported in the book by Ore [1].Over the years, diverse variants of the domination number in graphs have been proposed and studied.Some of the variants include the locating-dominating number, k-domination number and the Roman domination number, among others.There is a rich amount of the literature which has been published on the domination theory of graphs.For instance, Atakul [2] investigated the domination exponential and stability in certain families of graphs.Li [3] derived some upper/lower bounds on the Roman domination numbers of graphs.Hernández-Ortiz et al. [4] studied the weak Roman domination in certain infinite families of rooted product graphs.Cabrera-Martínez & Peiró [5] investigated the {k}-domination number of graphs with k = 2.For a detailed survey of the mathematical results on the domination theory of graphs, we refer the reader to Haynes et al. [6].
Well-covered graphs constitute an important class of graphs introduced by Ravindra [7] back in 1977.They have been extensively studied since then.For instance, Favaron [8] extended the notion of well-covered graphs and introduced very well-covered graphs.Finbow et al. [9] characterized well-covered graphs with a girth greater than or equal to 5. King [10] classified a subclass of well-covered networks.For the wellcoveredness of product graphs with such strong products of graphs, Topp & Volkmann [11] published some results.Recently, in 2018, Cartesian product graphs with well-covered properties were investigated by Hartnell et al. [12].
Well-dominated and k-extendable graphs form another structurally important class of graphs.In 1992, Anunchuen & Caccetta [13] introduced and studied critically k-extendable graphs.Well-dominated graphs first appeared in a seminal paper by Finbow et al. [14], who introduced them in the context of well-covered graphs.The well-dominated properties of a graph have an extensive amount of published results.Topp & Volkmann [15] in 1990 studied the well-coveredness and well-dominated properties of uniclycic as well as block graphs.Zverovich & Zverovich [16] proposed locally independent well-dominated and locally welldominated graphs and showed some structural results.Gözüpek et al. [17] investigated the well-dominated property of graphs obtained from the lexicographic product of two graphs.Graphs which are {C 4 , C 5 }-free have been studied in the context of well-dominated properties by Levit & Tankus [18].Alizadeh & Gözüpek [19] investigated bipartite graphs H, which are almost well-dominated having δ(H) ≥ 2 (the minimum degree in H of at least 2).For a survey on well-dominated graphs, we suggest the readers turn to a survey by Anderson et al. [20].Recently, in 2023, Rall [21] studied the well-dominated properties of strong, direct and Cartesian product graphs.
There have been numerous recent developments on well-dominated, well-covered and k-extendable graphs.For instance, in addition to introducing the classical well-dominated graphs, Anderson et al. [22] introduced the edge version of the well-dominated graphs.Kuenzel & Rall [23] in 2024 classified the well-dominated Cartesian product of graphs.In their seminal paper, Crupi et al. [24] investigated very well-covered graphs by algebraic structures such as Betti splittings.By putting certain conditions on ∆ (largest degree) and δ (smallest degree), Levit & Tankus [25] investigated well-covered graphs.Alves et al. [26] solved the graph sandwich problem in the context of partitions and well-coveredness.Regarding k-extendable graphs, Gan et al. [27] studied the k-extendability of Cayley graphs generated by transpositions.Feng et al. [28] investigated Hamiltonian cycle properties in k-extendable non-bipartite graphs with high connectivity.For implementation of machine learning by employing graph theory such as graph neural networks, we refer to [29][30][31][32][33].
In this paper, by extending the concept of k-extendable graphs, we introduce (k, γ)endowed graphs.We characterize minimal dominating sets in (k, γ)-endowed graphs and study their well-dominated property.We also introduce and study the symmetric γ-endowed graphs and prove some structurally important results.In this work, an open problem (from Khan [34]) asking to find a domination-dependent graphical descriptor delivering a correlation coefficient of ρ > 0.9967 with the total π-electronic energy E π of lower benzenoid hydrocarbons (BHs) has been solved.Section 6 shows that the upper domination number studied in this paper delivers a correlation coefficient ρ = 0.99905 > 0.9967 with E π of lower BHs.A regression model is also proposed with a detailed statistical analysis for structure-property studies of the E π of lower BHs.

Preliminaries
Consider a simple graph H = (V, E), where the number of vertices adjacent to a given vertex v ∈ V(H) is denoted by deg(v).The graph's minimum and maximum degrees are represented by δ(H) and ∆(H), respectively.The set of vertices adjacent to v in H is designated as N(v), and the closed neighborhood of v is defined as N[v] = N(v) ∪ {v}.The complement of a graph H, denoted as H, is characterized by vertices being adjacent only if they are nonadjacent in H.For a subgraph G of H, its vertex and edge sets are denoted as and y ∈ V 2 or vice versa.Various well-known graph structures are hereby identified, including the n-dimensional complete graph K n , cycle graph C n , path graph P n and the (m, n)-dimensional complete bipartite graph K m,n .Note that the start graph is merely the complete bipartite graph K 1,n .In a connected graph, it is noteworthy that every pair of vertices is connected by a path.A subset T ⊂ V(H) is said to be an independent set if no pair of vertices of T are adjacent in H.The maximum cardinality of an independent set is called the independence number β 0 (H) of H.For basic definitions, we refer the reader to [35,36].
A subset T within the vertex set V(H) is termed a dominating set (DS) of graph H if, for each vertex v in the complement of T there exists a vertex u in T such that u and v are adjacent.It is noteworthy that any superset of a dominating set of H also qualifies to be a dominating set of H [6,37] The smallest (resp.largest) cardinality of a minimal DS in graph H is termed as the domination number (γ(H)) (resp.upper domination number (Γ(H)).A DS is characterized as an independent DS if all the vertices in it are independent (i.e., mutually non-adjacent).The cardinality of the smallest independent DS is defined as the independent domination number and is represented by i(H).For instance, in the star graph K 1,n where n ≥ 2, the set of pendant vertices qualifies to be a DS, however, it does not encompass a minimal DS.
Berge [38] initiated the study of independent sets being contained in maximum independent sets.The natural curiosity is to start a similar study of DSs comprising a minimum DS.The minimum cardinality of such DSs are called γ-sets.Some dominating sets of a graph contain a minimum dominating sets while some may not.Claude Berge in 1980 defined k-extendable graphs as those in which any independent set of cardinality k is part of a maximum independent set of the graph.Graphs which are 1-extendable are called Berge graphs.This has led to the concepts of well-covered and well-dominated graphs.Analogous to k-extendable graphs, the concept of (k, γ)-endowed graphs is being introduced in this paper.
In (k, γ)-endowed graphs, every dominating set of cardinality k contains a minimum dominating set of the graph.Alternatively, a dominating set of a graph H containing a minimum dominating set of H is called a γ-endowed dominating set of H.If that set is of cardinality k then it is called a (k, γ)-endowed dominating set.If in a graph H, minimal dominating sets of cardinality greater than γ(H) exist, such sets will not qualify to be γendowed.Moreover if γ(H) = Γ(H), then every dominating set of H contains a minimum dominating set of H.This is one extreme.The other extreme is that for every positive integer γ(H) < k < Γ(H), there exists a minimal dominating set of H of cardinality k.The former concept is well-known and named as well-dominated graphs.In this article, we discuss the results related to (k, γ)-endowed graphs.
Definition 1.Let k be a positive integer.A simple graph H is called a (k, γ)-endowed graph if every dominating set of cardinality k in H contains a minimum dominating set of H.

1.
Note that every n-vertex graph possesses the (n, γ)-endowed property as well as the (γ, γ)endowed property.

3.
If the domination number γ(H) in graph H is less than its upper domination number Γ(H), then H does not exhibit the (Γ, γ)-endowed property.
We discuss the (k, γ)-endowed property of some standard graph families.
Let D r,s be the double star which is constructed by adjoining the centers of two stars K 1,s and K 1,r .Let r ≤ s.D r,s is (k, γ)-endowed for all k such that 2 ≤ k ≤ r and k = r + s + 2 and not (k, γ)-endowed for r In the case of the Peterson graph P,

9.
Assume that G = (C, I) is a complete split graph.Then, H is (k, γ)-endowed for all k except when l = |I|.
Remark 2. Let H = P n when n ≥ 5. We have γ(P n ) = ⌈ n 3 ⌉ (for any three consecutive vertices of P n , one vertex contributes to the DS), β 0 (P n ) = ⌈ n 2 ⌉ and i(P n ) = ⌊ n 2 ⌋.Since n ≥ 5, γ(P n ) < β 0 (P n ).Therefore, P n is not k-endowed when k = β 0 (H).If n ≥ 8, P n is not k-endowed when k = β 0 (P n ) and k = i(P n ).Moreover, It is not (n, γ)-endowed though.Note that the above example shows that a graph H may be (k, γ)-endowed for some positive integer k but may not be (l, γ)-endowed for some l > k.There exists a graph H in which H is (k, γ)-endowed but not (l, γ)-endowed for all l, where k < l ≤ n − 1. Figure 1 presents a graph H possessing γ(H) = 2.Moreover, H is not (k, γ)-endowed for any t, 3 ≤ t ≤ 5.

Open Problem:
Under what conditions on H, the property of being (k, γ)-endowed, is it implied that H is (l, γ)-endowed for all l, k ≤ l ≤ n?

1.
If H has a γ-fixed vertex u, i.e., that is every γ-set (a DS of cardinality γ) of H contains u which is not an isolated vertex, then H is not (k, γ)-endowed for all k, in which the existence of a DS of cardinality k is not containing u.In particular, H is not ((n − 1), γ)-endowed (when H is of order n).

2.
If H has t full degree vertices, then H is is the set of all full degree vertices then for any k such that γ(⟨H If H has a minimal DS of cardinality k > γ, then H is not (k, γ)-endowed.

Graphs with Unique Minimum DS
In this section, we study unique minimum dominating sets in (k, γ)-endowed graphs.Proof.Assume that u i lacks a pendant neighbor (pn) in the set V − T with respect to T. In this scenario, u i is an isolated member in the induced subgraph ⟨T⟩.As u i is not an isolated member in the original graph H, there must exist a vertex v ∈ N(u i ) ∩ (V − T).
Let T 1 = u 1 , u 2 , . . ., u i−1 , v, u i+1 , . . ., u k .Note that any u i 's neighbor (resp.the vertices u i ) is dominated by T 1 − {v} (resp.v).Consequently, T 1 qualifies as a minimum DS of H, leading to a contradiction.Hence, it is established that every vertex u i in T must have a pendant neighbor in V − T. Now, let us consider the case where u i possesses exactly one pendant neighbor in V − T concerning T. Denote the sole pendant neighbor of u i as v. Form the set T 2 = u 1 , u 2 , . . ., u i−1 , v, u i+1 , . . ., u k .In this configuration, v dominates u i , and any neighbor of u i (excluding v) is dominated by T 2 − {v}.Therefore, T 2 constitutes a minimum DS of H, resulting in a contradiction.
Consequently, it can be concluded that each vertex u i must have at least two pendant neighbors in V − T with respect to T.
Corollary 1.The induced graph ⟨pn(u i , T)⟩ has no full degree vertex for every u i ∈ T. Remark 4.

1.
Let H be a graph without isolated members.
Then, for any k 1 such that a ≤ k 1 ≤ n − k, there exists a DS of cardinality k 1 .These DSs do not contain T. Therefore, H is not There exists a graph with the unique minimum DS which is not ((γ + 2), γ)-endowed and which is ((γ + 1), γ)-endowed.For example, consider There exists a graph with a unique minimum DS which is (γ + 1, γ)-endowed and which is not Then, H has a unique minimum DS and γ(H Let H be a graph without isolated members.Let T = {u 1 , u 2 , . . ., u γ } be a unique minimum DS of H. Let min γ(⟨pn(u i , T)⟩) = a,.Then, H is not a (k, γ)-endowed for k satisfying Proposition 4. Assume a ∈ Z + to be the minimum satisfying γ(H) < a and H contains a minimal DS of cardinality a.Then, H is a (k, γ)-endowed for all k, γ(H) ≤ k ≤ a − 1.
Proof.Let T be a DS of cardinality k, γ(H) ≤ k ≤ a − 1, by hypothesis, T is not minimal and T cannot contain a minimal DS other than a γ-set.Therefore, T is a (k, γ)-endowed.
Corollary 2. Under the hypothesis of the above remark and assuming a ̸ = γ(H) + 1, H is not trivially γ-endowed.
Remark 5.There exists a graph H with a unique minimum DS such that every DS of cardinality γ(H) + 1 contains the unique minimum DS.
For the graph H in Figure 2, any DS of cardinality 5 contains T.There exists a DS of cardinality 6, namely {v 1 , v 2 , v 3 , v 14 , v 15 , v 16 } which does not contain T. Therefore, H is not a (6, γ)-endowed.In fact, H is not (k, γ)-endowed for any k such that 6 ≤ k ≤ 15.
Theorem 1. Suppose H is a graph without isolated members with a unique minimum DS T. Let ℓ = min Proof.Let T be a unique minimum DS of Therefore, γ(H) = 1.Proceeding as in case (i), we obtain the result.
Subcase (iii): ℓγ(H) Suppose T 1 is a DS of H of cardinality ℓ 1 ≤ γ + ℓ − 2. Suppose T 1 does not contain T.Then, for any vertex in V − T 1 , pn of that vertex belong to T 1 .Suppose T 1 does not contain t vertices of T, t ≥ 0. Therefore, H 1 contains 2tℓ vertices from V − T. Therefore, there exists a DS of cardinality ℓ 1 not containing a vertex u ∈ T for which |pn(u, T)| = ℓ.Hence, H is not (ℓ 1 , γ)-endowed for ℓ 1 with Proof.Since γ(D r,s ) = 2 and D r,s has a unique minimum DS T consisting of the two centers, say, u 1 and u 2 .pn(u 1 , T) = r if u 1 supports r pendant vertices.ℓ = min{pn(u 1 , T), pn(u 2 , Here we have a subsequent corollary to Theorem 2. Corollary 4. When k = 2, H is trivially γ-endowed.

Theorem 3. Let H be a graph without isolated members in which γ(H) < i(H) < β 0 (H).
Suppose there exists a minimum independent DS T in which at least one vertex, say u, has a pn in V − T with respect to T.Then, H is not (k, γ)-endowed for k = i and i + 1.
Proof.Suppose T is a minimum independent DS of H.Then, |T| = i ≥ 2.Moreover, by hypothesis, existence of a u ∈ T is ensured, satisfying pn(u, T) = ∅.Let v ∈ pn(u, T) and T 1 = T ∪ {v}.Suppose T 1 contains a minimum DS, say S of H. Clearly, S ̸ ⊂ T.
Since v ∈ pn(u, T), v can dominate at most one of u 1 , u 2 .Since S − {v} contains independent vertices of T, no vertices of S − {v} can dominate u 1 as well as u 2 , a contradiction, since S is a DS of H. Therefore, T 1 does not contain any minimum DS of H. Therefore, H is not ((i(H) + 1), γ)-endowed.Remark 8.

1.
Consider K m,n where m ≥ n ≥ 3, then γ(H) = 2 and i(H) = min{m, n} = n = 3.The partite set with n elements constitutes a minimum independent DS and no vertex of this set has a pn in the complement.

Proposition 5. Suppose H is a graph having no isolated members in which γ(H) < i(H).
Suppose there exists a minimum independent DS named Suppose S contains a γ-set of H, say, Remark 10.Existence of a graph H is depicted in Figure 4 with γ(H) < β 0 (H) and H is not (k, γ)-endowed for all k.We have the following: Case (ii): T is not independent.Let WLOG assume that u 1 ∼ u 2 .Then, u n is adjacent with exactly one of u 1 , u 2 ∈ T (since T is minimal).Therefore, Corollary 5.

1.
Consider a graph H with an order of n.The graph H possesses a minimal DS of cardinality n − 1 if and only if H can be expressed as Let H be a graph of order n with a minimal DS of cardinality n − 1.In such a case, H is trivially γ-endowed if and only if H takes the form K Example 4. Figure 5 presents a graph H with γ(H) = 2, Γ(H) = n − 2.  Theorem 5. Let u, v ∈ V(H) be such that any minimum DS of H either contains u or v. Suppose there exists a DS of cardinality γ + 1 not containing u and v.Then, H is almost trivially γ-endowed, that is, H is not (k, γ)-endowed for all k, γ(H) Proof.Let T be a DS of cardinality γ(H) + 1 not containing u and v.For any w not equal to u, v, we have that T ∪ {w} is a DS of cardinality γ(H) + 2 and it does not contain any minimum DS of H.
} is a DS of cardinality n − 2 and this does not contain any minimum DS.Thus, H is not (k, γ)-endowed for all k, γ(H) Corollary 6. Suppose u, v ∈ V(H) exist, satisfying the condition that any minimum DS of H either contains u or contains v and there exists a DS of cardinality γ(H) + 1 not containing u and v.Moreover, if H is not ((n + 1), γ)-endowed, then H is trivially γ-endowed.Consider C n , n ∼ = 1, 2( mod 3).Suppose n = 3t + 1 or 3t + 2.Then, γ(C n ) = t + 1 = n+2 3 or n+1 3 .Moreover, n is even.
For this class of graphs H, the graph H is (k, γ)-endowed for all k, γ(H)

Definition 2.
A graph H is said to be symmetric γ-endowed if for every k, γ(H) Let us illustrate this concept by studying it for standard families.Example 6.
Then, S ∪ {u} is a DS of cardinality γ(H) + 2. By hypothesis, S ∪ {u} is γ-endowed.There exists a γ-endowed set, say T 1 , which is contained in S ∪ {u} and hence contained in T.
As a converse to Lemma 1, we have: Lemma 2. Let H be a simple graph which is not well-dominated.Suppose H is (k, γ)-endowed for all k ≥ γ(H)

Application of γ in QSPR Models
This section investigates a significant applicability of the upper domination number within the context of QSPR studies of benzenoid hydrocarbons (BHs).
The total π-electron (E π ) energy in BHs that can be modelled through structureproperty relationships has emerged as an active research field.Notably, Luči'c et al. [39] demonstrated a close interrelation between product/sum-connectivity descriptors, which exhibit significant accuracy in predicting E π for BHs.Their study focused on a selection of 30 BHs as test molecules.Subsequently, their work was expanded by Luči'c et al. to encompass generalized versions of these connectivity descriptors denoted as χ α and χ s α , identifying optimal values (χ −0.2661 and χ s −0.5601 ) that offer superior predictive capabilities for E π in BHs.Furthermore, Hayat and coauthors [40] (and Hayat et al. [41]) investigated the estimation ability of valency-related (and distance-dependent) indices commonly observed in BHs.For insights into the effectiveness of eigenvalue-dependent spectral descriptors in E π prediction for BHs, readers are directed to [42,43].Additionally, recent advancements in QSPR models for chemical/physical characteristics in biomolecular networks as well as nano-structures have been discussed in [44][45][46][47][48].For the predictive potential of graphical indices for thermodynamic properties of benzenoid hydrocarbons, we refer the reader to [49][50][51][52].For the structure-property modeling of different chemical properties of a specific set of test molecules, the reader is referred to [53][54][55].
In a recent work, Khan [34] carried out a comparison analysis of seven dominationdependent graphical descriptors (excluding Γ), aiming to establish correlations with the E π of lower BHs.Among these parameters, the work by Khan [34] showcased that the paired-domination number γ p exhibits the most significant capability, showing a high correlation number of ρ = 0.9967.The following question was raised by concluding the investigation.
Problem 7. Find a domination-dependent graphical descriptor γ for which the correlation coefficient ρ(E π , γ) of lower BHs exceeds ρ > 0.9967?This section addresses Problem 7 and demonstrates that the upper domination number Γ yields an enhanced correlative capability, with E π of BHs exhibiting ρ(E π , Γ) = 0.99905 > 0.9967.To demonstrate this, 30 lower BHs as suggested by Khan [34] were opted.Figure 11 displays the BHs investigated in this analysis.
Subsequently, we determine the locating-dominating number Γ for the 30 BHs depicted in Figure 11.The values of the locating-dominating number Γ and E π (β), measured in units of β, are presented in Table 1 for the 30 lower BHs shown in Figure 11.Utilizing Table 1's data, we conducted thorough regression and correlation analyses.Initially, the correlation coefficient valuing ρ = 0.99905 was calculated, which is considerably higher than the minimum value of ρ = 0.9967.Subsequently, a comprehensive statistical data investigation was carried out.The most significant data-fitting regression model suggested by data analysis is linear.The subsequent details include a regression equation with 95% estimated confidence values and other important statistical values derived from Table 1.

Conclusions
The concept of (k, γ)-endowed graphs has been put forward.Furthermore, we characterized the graphs with unique minimum D-sets.In the last section, we introduced and studied symmetric γ-endowed graphs.In the second part of the paper, we found a domination-based parameter which showcases a correlation value of ρ = 0.99905 with E π (β) of lower BHs, thus answering an open problem in [34].The following problems are naturally raised from the findings of this study.

Proposition 2 .
Let H be a graph without isolated members.Let T = {u 1 , u 2 , • • • , u k } be a unique minimum DS of H. Every vertex u i ∈ T has at least two pn in V − T with respect to T. (That is, |pn(u, D) ∩ (V − T)| ≥ 2).

Figure 3 .
Figure 3.The graph H in Remark 8.

Figure 4 .Theorem 4 .
Figure 4.The graph H in Remark 10.Theorem 4. Assume an n-vertex graph H comprises a minimal DS of cardinality n − 1, then (i) γ(H) = n − 1 if H has exactly (n − 2) isolated members.(ii) γ(H) = n − t if H has exactly (n − t − 1) isolated members and the remaining vertices form a star.
isolated members and the remaining vertices form a star.Proof.Let T be a minimal DS of cardinality n − 1.Let γ(H) = {u 1 , u 2 , • • • , u n } and let T = {u 1 , u 2 , • • • , u n−3 }.Then, u n is adjacent to some point of D, say, u i , 1

Observation 2 .
If H is (k, γ)-endowed for all k, γ(H) ≤ k ≤ n except for exactly one value of k, say, l then in the sequence γ(H) ≤ i(G) ≤ β 0 (H) ≤ Γ(H), there are only two distinct values, as well as any minimal DS of H or a Γ-set (a minimum DS with cardinality Γ(H)) of H.The following two graphs satisfy the above property:(i) Let H = K 2,4 .Then, γ = 2, β 0 (H) = Γ(H) =4 and H is (k, γ)-endowed for all k, 2 ≤ k ≤ 6 except for k = 4. (ii) Assume the graph H is obtained from 2C 5 by joining one vertex of one C 5 with exactly one vertex of another C 5 .See Figure 7.

Figure 11 .
Figure 11.Graphical structures of the test molecules which have been considered.
. Moreover, a dominating set T of H is categorized as a minimal dominating set of H if no proper subset of T possesses the property of being a dominating set of H. Minimal DSs of a graph have been characterized by Ore in his famous Ore's theorem.A vertex z ∈ V(H) is said to be a private neighbour (pn) of y ∈ T ⊂ V(H) with respect to T denoted by pn[z, T] if z ∈ N[y] − N[T − {y}].A DS Tis minimal if and only if the private neighboured set of any vertex of T is non-empty.Note that any DS of H contains a minimal DS of H.
, C 3n (n ≥ 2) are not symmetric γ-endowed.Observation 3. A graph H is symmetric γ-endowed if and only H is well-dominated.Proof.Let H be well-dominated.Then, we haveΓ(H) = γ(H).Let T be any DS of H of cardinality k ≥ γ(H).Then, D contains a minimal DS of H. But, any minimal DS of H is a minimum DS of H. Therefore, H is (k, γ)-endowed.Therefore, H is symmetric γ-endowed.Conversely, if H is symmetric γ-endowed then Γ(H) = γ(H) and hence H is well-dominated.If n ∼ = 0( mod k) and n ≥ 12, ⌊ n 2 ⌋ − ⌈ n 3 ⌉ ≥2 and hence there exists a DS of cardinality ⌈ n 3 ⌉ + 1 which does not contain a minimum DS.If n ∼ = 1 or 2 (mod k) and n ≥ 16, then ⌊ n 2 ⌋ − ⌈ n 3 ⌉ ≥ 2. There exists a DS of cardinality ⌈ n 3 ⌉ + 1 which does not contain a minimum DS.It can be easily verified that C 6 , C 9 and C 12 are not γ-endowed.The same goes for C 8 , C 11 , and C 14 . Proof.