Excellent graphs with respect to domination: subgraphs induced by minimum dominating sets

A graph $G=(V,E)$ is $\gamma$-excellent if $V$ is a union of all $\gamma$-sets of $G$, where $\gamma$ stands for the domination number. Let $\mathcal{I}$ be a set of all mutually nonisomorphic graphs and $\emptyset \not= \mathcal{H} \subsetneq \mathcal{I}$. In this paper we initiate the study of the $\mathcal{H}$-$\gamma$-excellent graphs, which we define as follows. A graph $G$ is $\mathcal{H}$-$\gamma$-excellent if the following hold: (i) for every $H \in \mathcal{H}$ and for each $x \in V(G)$ there exists an induced subgraph $H_x$ of $G$ such that $H$ and $H_x$ are isomorphic, $x \in V(H_x)$ and $V(H_x)$ is a subset of some $\gamma$-set of $G$, and (ii) the vertex set of every induced subgraph $H$ of $G$, which is isomorphic to some element of $\mathcal{H}$, is a subset of some $\gamma$-set of $G$. For each of some well known graphs, including cycles, trees and some cartesian products of two graphs, we describe its largest set $\mathcal{H} \subsetneq \mathcal{I}$ for which the graph is $\mathcal{H}$-$\gamma$-excellent. Results on $\gamma$-excellent regular graphs and a generalized lexicographic product of graphs are presented. Several open problems and questions are posed.


Introduction
All graphs in this paper will be finite, simple, and undirected. We use [8] as a reference for terminology and notation which are not explicitly defined here. For a graph G = (V (G), E(G)), let π be a graphical property that can be possessed, or satisfied by the subsets of V . For example, being a maximal complete subgraph, a maximal independent set, acyclic, a closed/open neighborhood, a minimal dominating set, etc. Suppose that f π and F π are the associated graph invariants: the minimum and maximum cardinalities of a set with property π. Let µ ∈ {f π , F π }. For a graph G, denote by M µ (G) the family of all subsets of V (G) each of which has property π and cardinality µ(G). Each element of M µ (G) is called a µ-set of G. Fricke et al. [6] define a graph G to be µ-excellent if each its vertex belongs to some µ-set. Perhaps historically the first results on µ-excellent graphs were published by Berge [1] who introduced the class of B-graphs consisting of all graphs in which every vertex is in a maximum independent set. Of course all B-graphs form the class of β 0 -excellent graphs, where β 0 stand for the independence number. The study of excellent graphs with respect to the some domination related parameters was initiated by Fricke et al. [6] and continued e.g. in [3,9,10,14,18,20,23].
In this paper we focus on the following subclass of the class of µ-excellent graphs. Definition 1. Let I be a set of all mutually nonisomorphic graphs and ∅ = H I. We say that a graph G is H-µ-excellent if the following hold: (i) For each H ∈ H and for each x ∈ V (G) there exists an induced subgraph H x of G such that H and H x are isomorphic, x ∈ V (H x ) and V (H x ) is a subset of some µ-set of G. (ii) For each induced subgraph H of G, which is isomorphic to some element of H, there is a µ-set of G having V (H) as a subset.
By the above definition it immediately follows that each H-µ-excellent graph is µ-excellent. If a graph G is H-µ-excellent and H contains only one element, e.g. H = {H}, we sometimes omit the brackets and say that a graph G is Hµ-excellent. Define the µ-excellent family of induced subgraphs of a µ-excellent graph G, denoted by G µ , as the family of all graphs H ∈ I for which G is H-µ-excellent. The next two observations are obvious.
Observation 3. Let a graph G be both µ-excellent and ν-excellent. If the set of all µ-sets and the set of all ν-sets of G coincide, then G µ = G ν .
As first examples of H-µ-excellent graphs let us consider the case µ = β 0 . Clearly, any β 0 -excellent graph G is {K 1 , K β 0 (G) }-β 0 -excellent. A graph is rextendable if every independent set of size r is contained in a maximum independent set (Dean and Zito [4]). Clearly, a graph is {K 1 , K 2 , .., K r }-β 0 -excellent if and only if it is s-extendable for all s = 1, 2, .., r. Plummer [15] define a graph G to be well covered whenever G is k-extendable for every integer k. In other words, a graph G is well covered if and only if G β 0 = {K 1 , K 2 , .., K β 0 (G) }.
In this paper we concentrate mainly on excellent graphs with respect to the domination number γ. We give basic terminologies and notations in the rest of this section. In Section 2 we describe the γ-excellent family of induced subgraphs for some well known graphs. In Section 3 we show that, under appropriate restrictions, the generalized lexicographic product of graphs has the same excellent family of induced subgraphs with respect to six dominationrelated parameters. Section 4 contains results on γ-excellent regular graphs and trees. We conclude in Section 5 with some open problems.
In a graph G, for a subset S ⊆ V (G) the subgraph induced by S is the graph S with vertex set S and two vertices in S are adjacent if and only if they are adjacent in G. The complement G of G is the graph whose vertex set is V (G) and whose edges are the pairs of nonadjacent vertices of G. We write K n for the complete graph of order n and P n for the path on n vertrices. Let C m denote the cycle of length m. For any vertex x of a graph G, N G (x) denotes the set of all neighbors of x in G, N G [x] = N G (x) ∪ {x} and the degree of x is deg G (x) = |N G (x)|. The minimum and maximum degrees of a graph G are denoted by δ(G) and ∆(G), respectively. For a subset S ⊆ V (G), let A leaf is a vertex of degree one and a support vertex is a vertex adjacent to a leaf. The 1-corona, denoted cor(U), of a graph U is the graph obtained from U by adding a degreeone neighbor to every vertex of U. An isomorphism of graphs G and H is a bijection between the vertex sets of G and H f : V (G) → V (H) such that any two vertices u and v of G are adjacent in G if and only if f (u) and f (v) are adjacent in H. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G ≃ H. We use the notation [k] for {1, 2, .., k}.
An independent set is a set of vertices in a graph, no two of which are adjacent. The independence number of G, denoted β 0 (G), is the maximum size of an independent set in G. The independent domination number of G, denoted by i(G), is the minimum size of a maximal independent set in G. A subset D ⊆ V (G) is called a dominating set (or a total dominating set) in G, if for each x ∈ V (G) − D (or for each x ∈ V (G), respectively) there exists a vertex y ∈ D adjacent to x. A dominating set R of a graph G is a restrained dominating set (or an outer-connected dominating set) in G, if every vertex in V (G) − R is adjacent to a vertex in V (G) − R (or V (G) − R induces a connected graph, respectively). The minimum number of vertices of a dominating set in a graph G is the domination number γ(G) of G. Analogously the total domination number γ t (G), the restrained domination number γ r (G) and the outer-connected domination number γ oc (G) are defined. The minimum cardinality of a set S which is simultaneously total dominating and restrained dominating in G is called the total restrained domination number γ tr (G) of G. The minimum cardinality of a set S which is simultaneously total dominating and outer-connected dominating in G is called the total outer-connected domination number γ oc t (G) of G.

Examples
Here we find the γ-excellent family of induced subgraphs of some well known graphs.
Example 5. Let ν ∈ {γ, i}. Then all the following hold: The proof is straightforward and hence we omit it. From the above example, one can easily obtain the next result.
Example 6. Let a graph G be an union of s ≥ 2 paired disjoint cycles C n 1 , C n 2 , .., C ns .
Denote by (CEA) the class of all graphs G such that γ(G + e) = γ(G) for all e ∈ E(G).

Example 7. Let a noncomplete graph G be in (CEA). It is well known fact that any two nonadjacent vertices of G belong to some γ-set of G (Sumner and
Blitch [21]). In other words, G is {K 1 , K 2 }-γ-excellent graph.
Proof. Every independent set of G is a subset of a maximal independent set. Since each maximal independent set is always dominating and β 0 (G) = γ(G) = s, the result immediately follows.
The Cartesian product of two graphs G and H is the graph G H whose vertex set is the Cartesian product of the sets V (G) and V (H). Two vertices (u 1 , v 1 ) and (u 2 , v 2 ) are adjacent in G H precisely when either ., A m are γ-sets of G, and if m = n, B 1 , B 2 , .., B n are also γ-sets of G. Hence, by Proposition 8, G is {K 1 , K 2 , .., K m }-γ-excellent and Then there is a γ-set D of G such that D has an induced subgraph H 1 ≃ H. Assume that H has at least one edge.
Case 1: m < n. Clearly |A i ∩ D| = 1 for all i = 1, 2, .., m. Because of symmetry, we assume without loss of generality that D ∩ B j is empty for all j > m. Define now the set Hence H is either a complete graph or a union of complete and edgeless graph. Finally, it is easy to see that for each such a graph H, G is H-γ-excellent.
We need the following "negative result".
Proof. Assume that G is a P 3 -γ-excellent graph, γ(G) = 3 and x 1 , . But now no vertex of the induced path y 2 , y 3 , x 3 is adjacent to x 1 , a contradiction.
Proof. First note that K 3 K 3 ≃ K 3 K 3 and by Example 9 it immediately follows that Let us consider the graph G m,n = K m K n as a m × n array of vertices Remark now that: (a) {a i,j , a k,l , a r,s } ≃ K 3 if and only if both 3-tuples (i, k, r) and (j, l, s) consist of paired distinct integers. The vertices of each triangle of G m,n form a γ-set. Every two adjacent vertices a i,j and a k,l belong to a triangle.
(b) All induced subgraphs isomorphic to K 1 ∪ K 2 are {a i,j , a k,l , a i,l } and {a i,j , a k,l , a k,j } , where i = k and j = l. The vertices of each such a subgraph form a γ-set. Every two vertices belong to an induced subgraph isomorphic to K 1 ∪ K 2 . (c) Each 3-cardinality subset of Z j is independent and it is not dominating. Theorem 10 together with (a)-(c) immediately lead to the required.
To continue we need the following theorem and definitions.
. A subgraph of G H induced by a G-layer or an H-layer is isomorphic to G or H, respectively. Theorem 12. Let H be a connected noncomplete n-order graph and p ≥ n ≥ 3. If each induced subgraph of K p H which is isomorphic to H has as a vertex set some H-layer, then γ(K p H) = n and K p H is a H-γ-excellent graph.
Proof. Each H-layer of K p H is a dominating set of K n H. Hence γ(K p H) ≤ |V (H)| = n. Since p ≥ n, by Theorem A we have that each H-layer is a γ-set of K p H. It remains to note that clearly each vertex of K p H belongs to some H-layer.
The next example serves as an illustration of the above theorem.
Example 13. If p ≥ n ≥ 5, then the graph K p C n is C n -γ-excellent.
Proof. Let H be an induced subgraph of K p C n which is isomorphic to C r . It is easy to see that if H is not a C n -layer, then either r ∈ {3, 4} or r ≥ n + 2. The required immediately follows by Theorem 12.  j). The equality dist G[Φ] (x, y) = dist G (i, j) will be used in the sequel without specific references.

Generalized lexicographic product
Then clearly for each j ∈ N(i), V (F j ) ∩ D is empty and for any u j ∈ V (F j ) the set (D − {v 2 , .., v r }) ∪ {u j } is a dominating set of G[Φ] or a total dominating set of G[Φ] depending on whether µ = γ or µ = γ t , respectively. Hence r = 2. Since G is connected of order n ≥ 2 and |V (F i )| ≥ 3 for all i ∈ [n], the graph V (G[Φ]) − D is connected. Therefore the first two equality chains are correct.
Finally, let D 1 be a γ-set of G[Φ] and γ(F k ) ≥ 3 for all k ∈ [n]. Then clearly for every i ∈ [n] the sets D and V (F i ) must have no more than one element in common. But this immediately implies that D 1 is a total dominating set of G[Φ]. Thus, the last equality chain holds.

Theorem 15. Given a graph G[Φ]
, where G is connected of order n ≥ 2 and Proof. Recall that any G-layer of G[Φ] induces a graph isomorphic to G. We need the following claim.
. But then D is a dominating set of any subgraph of G[Φ] that is induced by a G-layer containing D. In particular this leads to If D * is a γ-set of some subgraph of G[Φ] that is induced by a G-layer, then again by the fact that all F i 's are complete, it follows that D * is a dominating set of G[Φ]. This clearly leads to γ(G[Φ]) ≥ γ(G).

Thus γ(G[Φ]) = γ(G) implying the required.
⇐ Choose u ∈ V (G[Φ]) arbitrarily. Then there is a G-layer U containing u. Since G is K s -γ-excellent, there is a γ-set D * of U that contains s paired nonadjacent vertices one of which is u. By Claim 1, D  *  is a γ-set of G[Φ].
If R is a s-vertex independent set in G[Φ], then since all F i 's are complete graphs, R is a subset of some G-layer. The rest is as above.
and a γ-set D of G[Φ] such that u ∈ I s ⊆ D. By Claim 1, D is a γ-set of some subgraph induced by a G-layer of G[Φ]. Since all F i 's are complete, without loss of generality, we can assume that D ⊆ L.
Let R be a s-vertex independent set of L. Then there is a γ-set D 1 of G[Φ] which has R as a subset. By Claim 1 D 1 is a γ-set of a graph induced by some G-layer and as above we can assume that D 1 ⊆ L.

Regular graphs and trees
To present the next results on regular graphs, we need the following theorem.
For any 5-regular graph G with γ(G) = 3, the bound stated in Theorem B can be improved by 3.
Proof. By Theorem B we have n ≥ 9. Since there is no 5-regular graphs of odd order, n ≥ 10 is even. Note that there are exactly sixty 5-regular graphs of order 10 [12,13]. Their adjacency lists can be found in [13]. A simple verification shows that each of these graphs has the domination number equals to 2.
Theorem 17. Let G be a s-regular K r -γ-excellent n-order connected graph with γ(G) = r, where n > s ≥ r ≥ 3. Then the following assertions hold.
(ii) If r = 3, then s ≥ 4 with equality if and only if n = 9 and G is one of the graphs depicted in Fig.1. (iii) If r = 3 and s = 5, then n = 12.
Proof. (i) Let H ≃ K r be a subgraph of G. Each vertex of H is adjacent to s − r + 1 vertices outside V (H). Hence n ≤ r + r(s − r + 1) = r(s − r + 2).
(ii) Since r = 3, we have γ(G) = 3 and n ≤ 3s − 3. By Theorem B we obtain 8 ≤ n when s = 3 and 9 ≤ n when s ≥ 4. Thus s ≥ 4 and if the equality Figure 1. The two 4-regular K 3 -γ-excellent graphs of order 9. The graph on the right is K 3 K 3 .
holds, then n = 9. There are exactly 16 4-regular graphs of order 9 [13]. An immediate verification shows that among them only the graphs depicted in Fig.1 are K 3 -γ-excellent.
Note that the connected 5-regular K 3 -γ-excellent graph depicted in Fig. 2 has order 12. Now we concentrate on graphs having cut-vertices. Let G 1 , G 2 , .., G k be pairwise disjoint connected graphs of order at least 2 and v i ∈ V (G i ), i = 1, 2, .., k. Then the coalescence (G 1 ·G 2 ·...·G k )(v 1 , v 2 , .., v k : v) of G 1 , G 2 , ..., G k via v 1 , v 2 , .., v k , is the graph obtained from the union of G 1 , G 2 , .., G k by identifying v 1 , v 2 , .., v k in a vertex labeled v. If for graphs To continue we need the following result: where H is connected and has no cut-vertex. Then G is also H-γ-excellent.
Proof. Using induction on k we easily obtain from Lemma C that {x} = Consider any induced subgraph R of G, which is isomorphic to H. Since H is connected and without cut-vertices, R is an induced subgraph of some G i , say without loss of generality, i = 1. Then there is a γ-set Define a vertex labeling of a tree T as a function S : V (T ) → {0, 1}. A labeled tree T is denoted by a pair (T, S). Let 0 T and 1 T be the sets of vertices assigned the values 0 and 1, respectively. In a labeled 1-corona tree of order at least four all its leaves are in 0 G and all its support vertices form 1 G .
Let T be the family of labeled trees (T, S) that can be obtained from a sequence of labeled trees τ : (T 1 , S 1 ), . . . , (T j , S j ), (j ≥ 1), such that (T 1 , S 1 ) is a labeled 1-corona tree of order at least four and (T, S) = (T j , S j ), and, if j ≥ 2, (T i+1 , S i+1 ) can be obtained recursively from (T i , S i ) by the following operation: Operation O. The labeled tree (T i+1 , S i+1 ) is obtained from vertex disjoint (T i , S i ) and a labeled 1-corona tree G i in such a way that Now we are in a position to present a (reformulated) constructive characterization of γ-excellent trees.
Theorem D. [17] For any tree T of order at least four the following are equivalent: (i) T is γ-excellent. Another constructive characterization of the γ-excellent trees can be found in [3]. To prove our last result we need the following lemma. Proof. Suppose T is H-γ-excellent where H is not edgeless. Let D be a γ-set of T and R ≃ H be an induced subgraph of D . Choose arbitrarily an edge xy of R. Clearly both x and y are not leaves and by Lemma 19, neither x nor y is a cut-vertex belonging to V − (T ). Hence x, y ∈ V = (T ), because of Theorem D. Now we choose xy so that x is a leaf in R. By Theorem D, a vertex y has a neighbor z ∈ V − (T ). Lemma 19 now implies N[z] ∩ D = {y}. But then the graph R x = V (R − x) ∪ {z} is isomorphic to R. Since z ∈ V − (T ) and yz ∈ E(T ), Lemma 19 shows that no γ-set of T contains both y and z.Thus, we arrive to a contradiction. Therefore, T γ contains only edgeless graphs. By Theorem D V − (T ) is a γ-set of T . Assume first that there is a cut-vertex x ∈ V − (T ). Then for any two neighbors y and z of x the set V 1 = (V − (T ) − {x}) ∪ {y, z} is independent of cardinality γ(T ) + 1. Suppose T is K r -γ-excellent for some r ≥ 2. Choose any cardinality r subset V 1 of (V − (T ) − {x}) ∪ {y, z} that contains both y and z. Now by Lemma 19, we conclude that no γ-set of T has V 1 as a subset. Thus, T γ = {K 1 }.
Finally, let V − (T ) contains only leaves. By Theorem D, T is a 1-corona tree. Clearly γ(T ) = i(T ) = β 0 (T ) = r and then the required now follows by Proposition 8.

Open problems and questions
We conclude the paper by listing some interesting problems and directions for further research.
• For which ordered pairs (r, s) there are s-regular K r -excellent graphs of order r(s − r + 2) (see Theorem 17)? Find all 12-order 5-regular K 3 -γ-excellent graphs. • Characterize/describe all graphs F such that there is no F -µ-excellent graph G with µ(G) = |V (F )| (see Observation 2). Recall that there is no P 3 -γ-excellent graph G with γ(G) = 3 (Theorem 10). • Let b be a positive integer. Denote by A (µ, b) the class of all µ-excellent connected graphs G for which µ(G) = b and |G µ | is maximum. It might be interesting for the reader to investigate these classes at least when b is small. Note that we already know that A (γ, 1) consists of all complete graphs, and all connected graphs obtained from K 2n , n ≥ 2, by removing a perfect matching form A (γ, 2) (Example 4). In addition, by Example 9 we have γ(K 3 K 3 ) = 3, K 3 K 3 γ = {K 1 , K 2 , K 2 , K 1 ∪ K 2 , K 3 , K 3 } and by Theorem 10 we know that there is no P 3 -γ-excellent graph G with γ(G) = 3. Thus, K 3 K 3 belongs to A (γ, 3) and |K 3 K 3 γ | = 6. Find A (γ, 3). • Find T µ for each µ-excellent tree T , where µ ∈ {i, γ t , γ R } and γ R stand for the Roman domination number (see [9], [10] and [18], respectively). • Find graphs H such that each induced subgraph of K p H which is isomorphic to H has as a vertex set some H-layer (see Theorem 12). • Characterize/describe all connected K 2 -γ-excellent graphs G with γ(G) = 2.