| Affiliation: | (1) Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, I.R., Iran;(2) Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA |
| Abstract: | The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If ![$$sum_{xin N[e]}f(x) geq 1$$](/content/lg01131623758274/373_2007_Article_762_TeX2GIFIEq1.gif) for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γs′(G) of G is defined as  . Recently, Xu proved that γs′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γs′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γs′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar. |
