Graphs with Constant Sum of Domination and Inverse Domination Numbers

A subsetD 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 inD. The domination number γ 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 toD. The inverse domination number γ ′ G is the minimum cardinality of the inverse dominating sets.Domke et al. 2004 characterized connected graphsGwith γ G γ ′ 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 Gwith minimum degree at least two and γ G γ ′ G n − 1.