On ve-degrees and ev-degrees in graphs

Let G = (V, E) be a graph with vertex set V and edge set E. A vertex v is an element of V ve-dominates every edge incident to it as well as every edge adjacent to these incident edges. The vertex edge degree of a vertex v is the number of edges ve-dominated by v. Similarly, an edge e = uv ev-dominates the two vertices u and v incident to it, as well as every vertex adjacent to u or v. The edge vertex degree of an edge e is the number of vertices ev-dominated by edge e. In this paper we introduce these types of degrees and study their properties. (C) 2016 Elsevier B.V. All rights reserved.