On The Strong Signed Domination Of Graph

Domination theory is an important branch of the graph theory, it has a wide range of applications in coding theory, computer science, network, society study and so on. Concurrent with the growth of computer science and the function method used, makes the domination theory become one of the fastest growing areas in recent decades of graph theory. The strong signed domination number of a graph has its important apply-ing background, so it is useful to investigate the lower bounds of it.In this paper, we pay our attention mainly on the following the four parts of dom-ination in graphs:In chapter1, we give some terminologies and notations, and some recent results concerning dominating number of graphs.In chapter2, we study of the strong signed domination number of a graph, point out the mistake of theorem5of reference [2] and improve the lower bounds of theorem4of reference [7], we obtain three sharp lower bounds of strong signed domination number separately, mainly by the following aspects:For any connected graph G of order n, if|E(G)|=m, then γss(G)≥[5n-2m/4], a sharp lower bound has been obtained as follows:C3∪Pn(2<n<5)∪Tn(2≤n≤5)∪K5.If G is connected graph of order n,|E(G)|=m,then γss(G)>≥2[1+(?)/2]—n, complete graph with even order is graph of sharp lower bound.For For any connected graph G of order n, if δ and Δ are minimum degree and maximum degree respectively, then γss(G)≥[(?)n], complete graph with even order is graph of sharp lower bound.In chapter3,we study Of the strong signed total domination number of a graph which is obtained by a modification of the signed domination number of a graph,we present the exact values of a cycle graph,a wheel graph,a complete graph,a complete bipartite graph on simple graph and five sharp lower bounds of the strong signed total domination number of a general graph, mainly by the following aspects:γsst(G)≥2[m+n/Δ]一n.γsst(G)≥2[(δ+2)n/δ+Δ]一n.γsst(G)≥2[(n-m)/δ]+n.γsst(G)≥2[4-δ+(?)/4]—n.γsst(G)≥2[2+(?)/4]—n.In chapter4,We obtain the minus cycle dominating number of a graph on the basis of the signed cycle domination number. We give bounds of a minus cycle dominating number on a general graph and determine the exact value of a minus cycle dominating number for some special classes of graphs,mainly by the following aspects:For an arbitrary graph G of order n,then γmc’.(G)≥1一n,with equality holding if and only if G is a tree.Let M={G1,G2,…,Gm,…),in which G1=K3,Gm is the graph obtained from Gm一1by adding a vertex u and three edges uv1. uv2and uv3,where v1,v2and v3are three vertices on the same inner face (a triangle)of Gm-1(i=1,2,3,…).For an arbitrary graph G of order n of M,then γmc’(G)=n一2.For an arbitrary positive integer2≤n, m, then γmc’(Km,m)=mn—m—n+1.