\(i\gamma(1)\)-Perfect Graphs with Girth at Least Six

Abstract

Let \(k\) be a nonnegative integer, and let \(\gamma(G)\) and \(i(G)\) denote the domination number and the independent domination number of a graph \(G\), respectively. The so-called \(i_k\)-perfect graphs consist of all such graphs \(G\) in which \(i(H) – \gamma(H) \leq k\) holds for every induced subgraph \(H\) of \(G\). This concept, introduced by I. Zverovich in \([5]\), generalizes the well-known domination perfect graphs. He conjectured that \(i\gamma (k)\)-perfect graphs also have a finite forbidden induced subgraphs characterization, as is the case for domination perfect graphs. Recently, Dohmen, Rautenbach, and Volkmann obtained such a characterization for all \(i\gamma(1)\)-perfect forests. In this paper, we characterize the \(i\gamma(1)\)-perfect graphs with girth at least six.