The edge-vertex domination and weighted edge-vertex domination problem

Abstract

Consider a simple (edge weighted) graph \(G = \left( {V,E} \right)\) with \(\left| V \right| = n\) and \(\left| E \right| = m\). Let \(xy \in E\). The domination of a vertex \(z \in V\) by an edge \(xy\) is defined as \(z\) belonging to the closed neighborhood of either \(x\) or \(y\). An edge set \(W\) is considered as an edge-vertex dominating set of \(G\) if each vertex of \(V\) is dominated by some edge of \(W\). The (weighted) edge-vertex domination problem aims to find an edge-vertex dominating set of \(G\) with the minimum cardinality. Let \(M \subseteq V\) and \(N \subseteq E\). Given a positive integer \(p\), if a vertex \(z\) is dominated by \(p\) edges in set \(N\), then set \(N\) is called a \(p\) edge-vertex dominating set of graph \(G\) with respect to \(M\). This study investigates the edge-vertex domination problem and the \(p\) edge-vertex domination problem, presents an algorithm with a time complexity of \(O\left( {nm^{2} } \right)\) for solving the weighted edge-vertex domination problem on unit interval graphs. Moreover, algorithms have been developed with time complexities of \(O\left( {m\lg m + p\left| M \right| + n} \right)\) and \(O\left( {n\left| M \right|} \right)\) for identifying a minimum \(p\) edge-vertex dominating set of an interval graph \(G\) and a tree \(T\), respectively, with respect to any subset \(M \subseteq V\).