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1	Vertex-Edge Domination Lewis, Jason, Hedetniemi, Stephen T., Haynes, Teresa W., Fricke, Gerd H. 01 March 2010 (has links) Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertexedge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number γve(G). We present bounds on γve(G) and relationships between γve(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that γve(G) is bounded above by the domination number of G. Our main result characterizes the trees having equal domination and vertex-edge domination numbers. domination total covering vertex-edge domination Mathematics and Statistics
2	On Ve-Degrees and Ev-Degrees in Graphs Chellali, Mustapha, Haynes, Teresa W., Hedetniemi, Stephen T., Lewis, Thomas M. 06 February 2017 (has links) Let G=(V,E) be a graph with vertex set V and edge set E. A vertex v∈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. degree edge–vertex degree edge–vertex domination irregular graph regular graph vertex vertex–edge degree vertex–edge domination Mathematics and Statistics

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Vertex-Edge Domination

On Ve-Degrees and Ev-Degrees in Graphs