Graphs with constant sum of domination and inverse domination numbers (Q446329)

Summary: A subset \(D\) of the vertex set of a graph \(G\) is a dominating set if every vertex in \(V - D\) is adjacent to at least one vertex in \(D\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A subset of \(V - D\), which is also a dominating set of \(G\) is called an inverse dominating set of \(G\) with respect to \(D\). The inverse domination number \(\gamma'(G)\) is the minimum cardinality of the inverse dominating sets. \textit{G.S. Domke} et al. [Ars Comb. 72, 149--160 (2004; Zbl 1077.05072)] characterized connected graphs \(G\) with \(\gamma(G) + \gamma'(G) = n\), where \(n\) is the number of vertices in \(G\). It is the purpose of this paper to give a complete characterization of graphs \(G\) with minimum degree at least two and \(\gamma(G) + \gamma'(G) = n - 1\).