Domination Graphs of Tournaments with Isolated Vertices

Abstract

The domination graph \(dom(D)\) of a digraph \(D\) has the same vertex set as \(D\), and \(\{u,v\}\) is an edge if and only if for every \(w\), either \((u,w)\) or \((v,w)\) is an arc of \(D\). In earlier work we have shown that if \(G\) is a domination graph of a tournament, then \(G\) is either a forest of caterpillars or an odd cycle with additional pendant vertices or isolated vertices. We have also earlier characterized those connected graphs and forests of non-trivial caterpillars that are domination graphs of tournaments. We complete the characterization of domination graphs of tournaments by describing domination graphs with isolated vertices.