Bipartite Theory of Graphs: Outer-Independent Domination

Abstract

Let \(G = (V,E)\) be a bipartite graph with partite sets \(X\) and \(Y\). Two vertices of \(X\) are \(X\)-adjacent if they have a common neighbor in \(Y\), and they are \(X\)-independent otherwise. A subset \(D \subseteq X\) is an \(X\)-outer-independent dominating set of \(G\) if every vertex of \(X \setminus D\) has an \(X\)-neighbor in \(D\), and all vertices of \(X \setminus D\) are pairwise \(X\)-independent. The \(X\)-outer-independent domination number of \(G\), denoted by \(\gamma _X^{oi}(G)\), is the minimum cardinality of an \(X\)-outer-independent dominating set of \(G\). We prove several properties and bounds on the number \(\gamma _X^{oi}(G)\).