Edge-vertex domination and total edge domination in trees

Abstract

An edge e is an element of E(G) dominates a vertex v is an element of V (G) if e is incident with v or e is incident with a vertex adjacent to v. An edge-vertex dominating set of a graph G is a set D of edges of G such that every vertex of G is edge-vertex dominated by an edge of D. The edge-vertex domination number of a graph G is the minimum cardinality of an edge-vertex dominating set of G. A subset D subset of E(G) is a total edge dominating set of G if every edge of G has a neighbor in D. The total edge domination number of G is the minimum cardinality of a total edge dominating set of G. We characterize all trees with total edge domination number equal to edge-vertex domination number.