In this paper we obtain some lower bounds for minus and signed domination numbers. We also prove and generalize a conjecture on the minus domination number for bipartite graph of order n, which was proposed by Jean Du...In this paper we obtain some lower bounds for minus and signed domination numbers. We also prove and generalize a conjecture on the minus domination number for bipartite graph of order n, which was proposed by Jean Dunbar et al [1].展开更多
Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G...Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G)→{−1,0,1}is said to be a mixed minus domination function(TMDF)of G if∑_(y∈Nt[x])f(y)≥1 holds for all x∈V(G)∪E(G).The mixed minus domination numberγ′_(tm)(G)of G is defined as γ′_(tm)(G)=min{∑x∈V∪E f(x)|f is a TMDF of G.In this paper,we obtain some lower bounds of the mixed minus domination number of G and give the exact values ofγ′_(tm)(G)when G is a cycle or a path.展开更多
基金Supported by the National Science Foundation of Jiangxi province(9911020).
文摘In this paper we obtain some lower bounds for minus and signed domination numbers. We also prove and generalize a conjecture on the minus domination number for bipartite graph of order n, which was proposed by Jean Dunbar et al [1].
基金This work was supported by the National Natural Science Foundation of China(No.11061014,11361024,11261019)the Science Foundation of Jiangxi Province(No.KJLD12067)The authors are grateful to the referees for their careful reading with corrections and especially the referee who draws our attention to the proof in Theorem 2.2,which let us improve the proof of Theorem 2.2,and correct this lower bound.
文摘Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G)→{−1,0,1}is said to be a mixed minus domination function(TMDF)of G if∑_(y∈Nt[x])f(y)≥1 holds for all x∈V(G)∪E(G).The mixed minus domination numberγ′_(tm)(G)of G is defined as γ′_(tm)(G)=min{∑x∈V∪E f(x)|f is a TMDF of G.In this paper,we obtain some lower bounds of the mixed minus domination number of G and give the exact values ofγ′_(tm)(G)when G is a cycle or a path.
