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 \(M \subseteq V\) .
