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ABTRACTFor a graph G, a subset of vertices D is a dominating set if for each vertex x not in D, x is adjacent to at least one vertex of D. The domination number, y(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T: trees, T, and T,, strict inequality occurs. We show Furthermore, w e supply a partial characterization for which pairs of for almost all pairs of trees.

~straetIn this paper we present a characterization of connected graphs of order 2n with domination number n. Using this class of graphs, we determine an infinite class of graphs with the property that the domination number of the product of any two is precisely the product of the domination numbers. IntroductionIn this paper, we discuss finite undirected simple graphs. For any undefined terms see [1]. Let G be a graph, we call V(G) the vertex set of 5 and denote by E(5) the edge set of 5. A subgraph H of 5, denoted H c_ (7, is a graph with V(H) c_ V(5) and E(H) c E(5). A subgraph H _ 5 is a spanning s~bgraph of 5 if V(H) -~ V(5). The product of two graphs, 5×H, is agraph with the following: V(5 × H) ----V(5) X V(H) and (gl,/h) (g2, h2) E E(5 X H) if and only if either gl ~ g2 and hlh ~ E E(H) or glg~ E E(5) and/h ~ ~. A subset S c_ V(G) is a dominating set of 5 if for every x E V(5) --S, there is at least one vertex y E S such that xy E E(5). Finally, the domi~at~ ~umber of a graph 5, denoted a(5), is the order of the smallest dominating set, There has been a good deal of research done on the domination number of a graph, see [2], and [3]. It is the purpose of this article to characterize connected graphs 5 of order 2n with a(5) = n. We go on to show that this class of graphs has the property that a(GxH)= a(G)a(H) for every two graphs in this class. Main resultsBefore proceding we present a few observations. Throughout the course of this paper we will use Ix] ([xJ) to denote the smallest (largest) integer greater (less) than or equal to x. Also a vertex v E V(G) is called an end-vertex AMS (MOS) subject classi/ication (1980). Primary 05C35.

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