Vertex-edge domination dot critical graphs

A vertex [Formula: see text] of a graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a vertex-edge dominating set (ved-set) if every edge of [Formula: see text] is ve-dominated by at least one vertex in [Formula: see text] The ve-domination number is the minimum cardinality of a ved-set of [Formula: see text]. We first show that the contraction of any edge of a graph [Formula: see text] can decrease the ve-domination number of [Formula: see text] by at most one. Then we mainly provide a constructive characterization of ve-dot-critical trees, that is, the trees whose ve-domination number decreases by the contraction of any edge.