Domination-compliance graphs are planar (Q2716664)

A tournament \(T\) is considered. Two vertices \(x\), \(y\) of \(T\) are a dominating (or compliant) pair, if for each vertex \(z\) distinct from both \(x\) and \(y\) either \(x\), or \(y\) beats \(z\) (or is beaten by \(z\), respectively). The domination graph \(\text{dom}(T)\) (or compliance graph \(\text{com}(T)\)) of \(T\) is an undirected graph whose vertex set is equal to that of \(T\) and in which two vertices are adjacent if and only if they form a dominating (or, respectively, compliant) pair in \(T\). The domination-compliance graph \(\text{DC}(T)\) of \(T\) is the union \(\text{dom}(T)\cup \text{com}(T)\). The structure of \(\text{DC}(T)\) is studied. The main result states that if \(T\) is a tournament whose domination graph is made up of nontrivial components, then \(\text{DC}(T)\) is planar.