一类联图的符号控制数

The Signed Domination Number of a Join Graph

设图G=(V,E)为一个图,一个双值函数f:V→{1,-1},若SV,则记f(S)=∑v∈Sf(v)。如果对任意的顶点v∈V,均有f(N[v])≥1成立,则称f为图G的一个符号控制函数,图的符号控制数定义为γs(G)=min{f(V)|f是图G的一个符号控制函数}图G=Km∨Pn表示图Km的每个顶点与路Pn的每个顶点相连接的联图。本文主要用分类讨论法和穷标法得到了图G的符号控制数的精确值,即确定了γs(Km∨Pn)。特别地,此联图中当m=1时得到了扇图的符号控制数,从而推广了已知结果。

Let G =( V,E) be a graph and denotes f(S)=∑ v∈S f(v) for S V . A function f:V→{1,-1} is said to be a signed domination function(SDF),if f(N[v])≥1 for v∈V . The signed domination number is γ s(G)= min {f(V)|f is an SDF of G . G= K m ∨P n denote is a graph obtained by every vertex of K m is adjacent to each vertex of path P n . In this paper, we determine exact values of the signed domination number of a join graph by exhaustive method and classified discussion, which is γ s( K m ∨P n). Specially, we give the signed domination number of Fan graph for m=1 , which generalize the known result.