Vizing-like Conjecture for the Upper Domination of Cartesian Products of Graphs - The Proof

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper domination numbers, in symbols: Γ(G H) ≥ Γ(G)Γ(H). A conjecture posed by Vizing [7] in 1968 claims that Vizing’s conjecture: For any graphs G and H, γ(G H) ≥ γ(G)γ(H), where γ, as usual, denotes the domination number of a graph, and G H is the Cartesian product of graphs G and H . It became one of the main problems of graph domination, cf. surveys [2] and [4, Section 8.6], and two recent papers [1, 6]. The unability of proving or disproving it lead authors to pose different variations of the original problem. Several such variations were studied by Nowakowski and Rall in the paper [5] from 1996. In particular, they proposed the following Conjecture (Nowakowski, Rall): For any graphs G and H, Γ(G H) ≥ Γ(G)Γ(H), where Γ denotes the upper domination of a graph. In this note we prove this conjecture. In fact, if both graphs G and H are nontrivial (i.e. have at least two vertices) we prove the following slightly stronger bound: Γ(G H) ≥ Γ(G)Γ(H) + 1. ∗Supported by the Ministry of Education, Science and Sport of Slovenia under the grant Z1-30730101-01. the electronic journal of combinatorics 12 (2005), #N12 1 We start with basic definitions. For graphs G and H , the Cartesian product G H is the graph with vertex set V (G) × V (H) where two vertices (u1, v1) and (u2, v2) are adjacent if and only if either u1 = u2 and v1v2 ∈ E(H) or v1 = v2 and u1u2 ∈ E(G). For a set of vertices S ⊆ V (G) × V (H) let pG(S), pH(S) denote the natural projections of S to V (G) and V (H), respectively. A set S ⊂ V (G) of vertices in a graph G is called dominating if for every vertex v ∈ V (G) \S there exists a vertex u ∈ S that is adjacent to v. A dominating set S is minimal dominating set if no proper subset of S is dominating. Minimal dominating sets give rise to our central definition. The upper domination number Γ(G) of a graph G is the maximum cardinality of a minimal dominating set in G. Recall that the domination number γ(G) is the minimum cardinality of a (minimal) dominating set in G. The following fundamental result due to Ore, cf. [3, Theorem 1.1], characterizes minimal dominating sets in graphs. Theorem 1 A dominating set S is a minimal dominating set if and only if for every vertex u ∈ S one of the following two conditions holds: (i) u is not adjacent to any vertex of S, (ii) there exists a vertex v ∈ V (G) \ S such that u is the only neighbor of v from S. Based on Ore’s theorem we present a partition of the vertex set of a graph depending on a given minimal dominating set. Let DG be a minimal dominating set of a graph G. If for a vertex u ∈ DG the condition (ii) of Theorem 1 holds, then we say that v is a private neighbor of u (that is, v is adjacent only to u among vertices of DG). Note that u can have more than one private neighbor. Also note that for a vertex u of DG both conditions of Theorem 1 can hold at the same time, that is, it can have a private neighbor and be nonadjacent to all other vertices of DG. Denote by D ′ G the set vertices of DG that have a private neighbor, and by PG the set of vertices of V (G) \ DG which are private neighbors of some vertex of D′ G. By NG we denote the set of vertices of V (G) \DG which are adjacent to a vertex of D′ G but are not private neighbors of any vertex of D ′ G. Set D′′ G = DG \D′ G denoting the vertices of DG which do not have private neighbors (so they must enjoy condition (i) of the theorem), and finally let the remaining set be RG, that is RG = V (G) \ (DG ∪PG ∪NG). We will skip the indices if the graph G will be understood from the context. Note that given a minimal dominating set D of a graph G, the sets D′, D′′, P, N and R form a partition of the vertex set V (G). In addition, some pairs of sets must clearly have adjacent vertices (like D′ and P ), while some other pairs of sets clearly do not have any adjacent vertices (like D′ and D′′). The situation is presented in Figure 1, where doubled line indicates that between two sets there must be edges, a normal line indicates that between the two sets edges are possible (but are not necessary), and no line between two sets means no edges are possible. Note that every vertex of R is adjacent to a vertex of D′′, and that every vertex of N ∪ P is adjacent to a vertex of D′. Of course, some of the sets could also be empty for some dominating sets. If A and B are two subsets of the vertex set of a graph we say that A dominates (vertices of) B if every vertex of B has a neighbor in A or is a vertex of A. We may then also say that B is dominated by (vertices of) A. the electronic journal of combinatorics 12 (2005), #N12 2 &% '$ &% '$ &% '$ &% '$ &% '$