VERTEX-EDGE ROMAN DOMINATION

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V, E) is a function f : V (G) -> {0, 1, 2} such that for each edge e = uv either max{f(u), f(v)} not equal 0 or there exists a vertex w such that either wu is an element of E or wv is an element of E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by gamma(veR)(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number gamma(veR)(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then gamma(veR)(T) >= (n - l - s + 3)/2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with gamma(veR)(T) = 2 gamma'(T), where gamma'(T) is the edge domination number of T.