Graphs with unique minimum edge-vertex dominating sets

An edge e of a simple graph G = (VG, EG) is said to ev-dominate a vertex v & ISIN; VG if e is incident with v or e is incident with a vertex adjacent to v. A subset D & SUBE; EG is an edge-vertex dominating set (or an evd-set for short) of G if every vertex of G is ev-dominated by an edge of D. The edge-vertex domination number of G is the minimum cardinality of an evd-set of G. In this paper, we initiate the study of the graphs with unique minimum evd-sets that we will call UEVD-graphs. We first present some basic properties of UEVD-graphs, and then we characterize UEVD-trees by equivalent conditions as well as by a constructive method.