A NEW LOWER BOUND ON TOTAL VERTEX-EDGE DOMINATION NUMBER OF A TREE

A vertex v of a graph G = (V, E) is said to vertex-edge dominate every edge incident to v, as well as every edge adjacent to these incident edges. A subset S & SUBE; V is a vertex-edge dominating set (ve-dominating set) if every edge of E is vertex-edge dominated by at least one vertex of S. A subset D & SUBE; V(G) is a total dominating set of G if every vertex of V(G) has at least one neighbor in D. The minimum cardinality of total dominating set of G is called total domination number-yt(G). A ve-dominating set is a total ve-dominating set if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of G is the total vertex-edge domination number-yvte(G). In this paper, we prove that-yvte(G)& GE; (-yt(T)- s+ 2)/2 for every non-trivial tree with s support vertices, and characterize extremal trees attaining this bound.