Well ev-covered trees

Abstract

An edge in a graph G = (V, E) is said to ev-dominate the vertices incident to it as well as the vertices adjacent to these incident vertices. A subset F ⊆ E is an edge-vertex dominating set (or simply, ev-dominating set) if every vertex is ev-dominated by at least one edge of F. The ev-domination number γev(G) is the minimum cardinality of a ev-dominating set of G. An ev-dominating set is independent if its edges are independent. The independent ev-domination number iev(G) is the minimum cardinality of an independent ev-dominating set and the upper independent ev-domination number βev(G) is the maximum cardinality of a minimal independent ev-dominating set of G. In this paper, we show that for every nontrivial tree T, γev(T) = iev(T) ≤ γ(T) ≤ βev(T), where γ(T) is the domination number of T. Moreover, we provide a characterization of all trees T with iev(T) = βev(T), which we call well ev-covered trees, as well as a characterization of all trees T with γev(T) = iev(T) = γ(T).