Minus Edge Domination Numbers Of Trees

The domination numbers of graphs play an important role in the structure of graphs.There are wide range of applications in coding theory,computer science,communication networks, surveillance systems,social networks,and other fields.Dunbar first introduced the signed domination number of graph.In 1998 Haynes published two monographs,in which a system overview of the domination theory is given,including signed domination number of graphs and minus domination number of graphs.These studies concentrate on vertices of graphs.Xu Bao-gen recently introduced edge domination numbers of graphs,and obtained some important results.Xu studied the minus edge domination numberγ'_m(G) of a graph and obtained two lower bounds of minus edge domination numbers,and determined minus edge domination numbers for complete graph,cycles and wheels.Liao Wansheng continued to investigate minus edge domination numbers.The best upper bound of minus edge domination number and two lower bound of minus edge domination numbers are given,minus edge domination numbers for some special graphs are determined.In this paper,we continue to investigate minus edge domination numbers of trees.The best lower bound of minus edge domination number of trees and upper bound of minus edge domination number of trees are given.Whenγ'_m(T)=1,its sufficient and necessary condition is given,and its structure is determined.Whenγ'_m(T)=2,its necessary condition is given.