Dominating direct products of graphs (Q879339)

Let \(G=(V,E)\) be a graph. A set \(S\subset V\) is called dominating if each vertex in \(V\backslash S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) of a graph \(G\) is the minimum cardinality of a dominating set. For graphs \(G\) and \(H\), the direct product \(G\times H\) is the graph with vertex set \(V(G)\times V(H)\) where two vertices \((x,y)\) and \((v,w)\) are adjacent if and only if \(xv\in E(G)\) and \(yw\in E(H)\). The purpose of the paper is to estimate the domination number and some related parameters (upper domination number and paired-domination number) of a direct product of graphs. Two of the main results are: (1) \(\gamma(G\times H)\leq 3\gamma(G)\gamma(H)\). (2) Graphs with arbitrarily large domination numbers for which there is an equality in this inequality are constructed.