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 ∈ vg if e is incident with v or e is incident with a vertex adjacent to v. a subset d ⊆ 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.