Signed Domination And Its Extention In Graph

Within the last forty years, the development of graph domination theory shows very quickly trend. and has a broad and important applications in various disciplines and real life. Then, it has attracted more and more scholars’concerning with exploring its content system and increasing rich graph of domination theory, however, it remains to be completed. This paper mainly studies the graph of the signed domination and its various forms of an extension, the main research results are:Firstly, we focus on several classes of signed domination, including graph of signed domination, k reverse signed domination, signed cycle (vertex) domination, and signed path (vertex) domination. We get some new lower bounds of the signed domination number γs(G) of a graph G, characterizing all graphs G satisfying γs (G)=|V(G)|, gaining a upper bound of the k reverse signed domination number of a graph, geting a lower bound for signed cycle (vertex) domination number γsc(G) of every maximal planar graph G, gaining a feature for all connected graphs of γsc(G)=|V(G)|-2, geting some signed cycle domination number for the graph G=P2×Pn,P3×Pn,P4×Pn, obtaining a new lower bound for signed path(vertex) domination number γ p (G) of every tree T.Secondly, we discusse edge signed domination of a graph, including reverse signed edge domination, k signed edge domination, and k reverse signed edge domination. We get some new upper bounds of reverse signed edge domination number of a graph, obtaining some new lower bounds of k signed edge domination of a graph, gaining the exact value of k signed edge domination for path and cycle, and related method conclusion is extended to k reverse signed edge domination in graphs.Lastly, we extensional edge of the signed domination, studying graph of signed star domination, reverse signed star domination, signed stark domination, and reverse signed star k domination. We get the signed star domination numbers for G=Pm×Pn, Pm×Cn and signed star k domination numbers for some special graphs. in addition, gaining a upper bound of the reverse signed star k domination number ofgraph.