Several Kinds Of K-subdomination Numbersof Graphs

In this paper, we mainly consider the k-subdomination for several kinds of graphs, thiscontents contain the following four aspects:（1） vertex signed domination number and vertex signed k-subdomination number;（2） signed edge domination number and signed edge k-subdomination number;（3） the signed edge total k-subdomination number and the reverse signed edge totalk-subdomination number of graphs;（4） the signed star k-subdomination number and the reverse signed star k-subdominationnumber of graphs.In Chapter2, we summarize and research mainly on the concept and charact of vertexsigned domination number and vertex signed k-subdomination number, and discuss thebounds on domination number for general graphs, also determined the exact values ofdomination number for the other special graphs.In Chapter3, first on the basis of the signed edge total domination, we introduce theconcept of the signed edge totalk-subdomination number, and give the two bounds of thesigned edge totalk-subdomination number of graphs, and also determined the exact values ofthe signed edge totalk-subdomination number for the special graphsPn,C nandSn1.And then on the basis of the reverse signed edge total domination, we introduce the conceptof the reverse signed edge totalk-subdomination number, given the two bounds of the reversesigned edge totalk-subdomination number of graphs, and also determined the exact values ofthe reverse signed edge total k-subdomination number for the special graphsPn andC n.Obtain the flowing results:I.The lower bounds on the signed edge total k-subdomination number of graphs1.For all graphs G （n≧3）, if the edge degrees sequenced d’₁≦d’₂≦…≦d’_m2.For all graphs G of order n, E （G） m,（2）and for the most degree andthe minimum degree of graphs respectively, thenII. The lower bounds on the signed edge total k-subdomination number of the special graphs1. Let n3be an integer, then 2. Let n≧3be an integer, then3. Let n≧3be an integer, thenIII. The lower bounds on the reverse signed edge total k-subdomination number of thespecial graphs1.For all graphs G （n3）, if the edge degrees sequenced d’₁≦d’₂≦…≦d’_m2.For all graphs G of order n, E （G） m,（2）and for the most degree andⅢthe minimum degree of graphs respectively, thenIn Chapter4, first we summarize and research mainly on the concept and charact of thesigned star k-subdomination number and discuss the bounds on domination number forgeneral graphs, also determined the exact values of domination number for the other specialgraphs. Then on the basis, obtained the concept the reverse signed star k-subdominationnumber, and two lower bounds of the reverse signed star k-subdomination number.IV.The lower bounds on the reverse signed star k-subdomination number of graphs1.For all graphs G of order n, if1,1k n1, then2.For all graphs G of order n, if1E （G） m, then V. The reverse signed star k-subdomination number of the special graphs1.Let n≧3be an integer, then（1）γ_rss（p_n）=n-2k-1;（2）γ_rss（c_n）=n-2k-1;...