Note: A note on the signed edge domination number in graphs

  • Authors:

  • Xueliang Fu;Yuansheng Yang;Baoqi Jiang

  • Affiliations:

  • Department of Computer Science, Dalian University of Technology, Dalian, 116024, PR China and College of Computer and Information Engineering, Inner Mongolia Agriculture University, Huhehote, 0100 ...;Department of Computer Science, Dalian University of Technology, Dalian, 116024, PR China;Department of Computer Science, Dalian University of Technology, Dalian, 116024, PR China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

Let G=(V(G),E(G)) be a graph. A function f:E(G)-{+1,-1} is called the signed edge domination function (SEDF) of G if @?"e"^"'"@?"N"["e"]f(e^')=1 for every e@?E(G). The signed edge domination number of G is defined as @c"s^'(G)=min{@?"e"@?"E"("G")f(e)|f is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541-1546] researched on the edge domination in graphs and proved that @c"s^'(G)@?@?116n-1@? for any graph G of order n(n=4). In the article, he conjectured that: For any 2-connected graph G of order n(n=2), @c"s^'(G)=1. In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G"m","k with @c"s^'(G"m","k)=-k(m-1)(km+k+1)2(k=2,m=1).