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On Well-Dominated Graphs On well-dominated graphs 1Sarah E. Anderson, 2Kirsti Kuenzel and 3Douglas F. Rall 1Department of Mathematics University of St. Thomas St. Paul, Minnesota USA Email: [email protected] 2Department of Mathematics Trinity College Hartford, Connecticut USA Email: [email protected] 3Department of Mathematics Furman University Greenville, SC, USA Email: [email protected] Abstract A graph is well-dominated if all of its minimal dominating sets have the same cardi- nality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs G or H has no isolatable vertices, we prove that the direct product of G and H is well-dominated if and only if either G = H = K3 or G = K2 and H is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2. Keywords: well-dominated, well-covered, Cartesian product, direct product, disjunctive product AMS subject classification: 05C69, 05C76 1 Introduction Given a graph G and a positive integer k, deciding whether G has a dominating set of cardinality at most k is one of the classic NP-complete problems [5]. If we restrict the input 1 graph to come from the class of well-dominated graphs (those graphs for which every minimal dominating set has the same cardinality), then this decision problem is solvable in linear time. The following simple procedure can be used to compute the order of a minimum dominating set in a well-dominated graph G. Let D = V (G). Choose any linear ordering of V (G) and process the vertices one at a time in this order. When a vertex v is processed, replace D by D − fvg if D − fvg dominates G. When all vertices in the sequence have been checked the resulting set D is a minimal (and hence minimum) dominating set of G. Since every maximal independent set of an arbitrary graph is also a minimal dominating set, one could also compute the domination number of a well-dominated graph by using a greedy algorithm to find a maximal independent set. The study of well-dominated graphs was initiated by Finbow, Hartnell and Nowakowski [4], and in that seminal paper they determined the well-dominated bipartite graphs as well as the structure of well-dominated graphs with no cycle of length less than 5. In [14] Topp and Volkmann gave a characterization of well-dominated block graphs and unicyclic graphs. The 4-connected, 4-regular, claw-free, well-dominated graphs were characterized by Gionet, King and Sha [6]. In more recent work, Levit and Tankus [11] proved that a graph with no cycles of length 4 or 5 is well-dominated if and only if it is well-covered. G¨oz¨upek, Hujdurovi´cand Milaniˇc[7] determined the structure of the well-dominated lexicographic product graphs. In this paper we first prove that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. In particular, we show that G is such a graph if and only if G is one of K1, K2, P4, C4, C5, C7, the corona of P3, or one of the four graphs in Figure 2. The main focus of the paper is on well-dominated graph products, and we investigate the Cartesian, direct and disjunctive products. The two main results concerning well-dominated Cartesian products are the following, the first of which has no restrictions on the factors while the second requires the two factors to be triangle-free. Theorem 1. Let G and H be connected graphs. If the Cartesian product G 2 H is well- dominated, then G or H is well-dominated. Theorem 2. Let G and H be nontrivial, connected graphs both of which have girth at least 4. The Cartesian product G 2 H is well-dominated if and only if G = H = K2. In Section 5 we determine all well-dominated direct products if at least one of the fac- tors does not have a vertex that can be isolated by removing from the graph the closed neighborhood of some independent set of vertices. Theorem 3. Let G and H be nontrivial connected graphs such that at least one of G or H has no isolatable vertices. The direct product G × H is well-dominated if and only if G = H = K3 or at least one of the factors is K2 and the other factor is a 4-cycle or the corona of a connected graph. Our main result concerning disjunctive products provides a complete characterization of well-dominated disjunctive products of two connected graphs of order at least 2. 2 Theorem 4. Let G and H be nontrivial connected graphs. The disjunctive product G _ H is well-dominated if and only if at least one of G or H is a complete graph and the other factor is well-dominated with domination number at most 2. The paper is organized as follows. In the next section we give the main definitions and also give some results about well-covered and well-dominated graphs that will be needed in the last four sections. In Section 3 we provide a characterization of the finite list of connected triangle-free graphs that are well-dominated and have domination number less than 4. In Section 4 we investigate well-dominated Cartesian products and prove Theorems 1 and 2. Sections 5 and 6 are devoted to proving Theorem 3 and Theorem 4 respectively. 2 Preliminaries Let G be a finite, simple graph with vertex set V (G) and edge set E(G). For a positive integer n we always assume the vertex set of the complete graph Kn is the set [n], which is defined to be the set of positive integers less than or equal to n. The girth of G is the length of its shortest cycle. An edge incident with a vertex of degree 1 is a pendant edge. A vertex x of G is isolatable if there exists an independent set I in G such that x is an isolated vertex in G − N[I]. Note that a vertex of degree 1 that is incident with a pendant edge is isolatable in G unless the pendant edge is a component of G. The distance in G between vertices u and v is the length of a shortest u; v-path in G and is denoted dG(u; v), or d(u; v) when the context is clear. We investigate graphs that arise as a Cartesian product X 2 Y , a direct product X × Y , or a disjunctive product X _ Y of smaller (factor) graphs X and Y . In all three of these graph products the vertex set is V (X) × V (Y ). The edge sets are defined as follows: 1. E(X 2 Y ) = f(x1; y1)(x2; y2):(x1 = x2 and y1y2 2 E(Y )) or (y1 = y2 and x1x2 2 E(X))g. 2. E(X × Y ) = f(x1; y1)(x2; y2): x1x2 2 E(X) and y1y2 2 E(Y )g. 3. E(X _ Y ) = f(x1; y1)(x2; y2): x1x2 2 E(X) or y1y2 2 E(Y )g. Suppose x 2 V (X) and y 2 V (Y ). If ∗ 2 f 2 ; ×; _g, then the X-layer of X ∗ Y containing (x; y) is the subgraph of X ∗ Y induced by the set V (X) × fyg and the Y -layer containing (x; y) is the subgraph induced by fxg × V (Y ). Note that in the Cartesian product or the disjunctive product, an X-layer is isomorphic to X while in the direct product an X-layer is a totally disconnected graph of order jV (X)j. A similar statement holds for a Y -layer. For x 2 V (G) the open neighborhood of x is the set N(x) consisting of all vertices in G that are adjacent to x and its closed neighborhood is the set defined by N[x] = N(x) [ fxg. Let S ⊆ V (G). The open (closed) neighborhood of S is the set N(S)(N[S]) defined as the union of the open (closed) neighborhoods of the vertices in S. For v 2 S, the closed private neighborhood of v with respect to S is the set pn[v; S] defined by pn[v; S] = fu 2 V (G): 3 N[u] \ S = fvgg. If pn[v; S] is nonempty, then each vertex in pn[v; S] is called a private neighbor of v with respect to S. A set D ⊆ V (G) is a dominating set of G if N[D] = V (G), and we then say that D dominates G. If fxg dominates G, then x is called a universal vertex. A dominating set D is minimal with respect to set inclusion if and only if pn[u; D] 6= ; for every u 2 D. The smallest cardinality among the minimal dominating sets is called the domination number of G and is denoted γ(G). The upper domination number of G is the number Γ(G), which is the largest cardinality of a minimal dominating set.
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