Some Notes on Signed Edge Domination in Graphs

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 $$\sum_{x\in N[e]}f(x) \geq 1$$ 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 $$\gamma_s^\prime(G) = {\text{min}}\{\sum_{e\in E(G)}f(e)\mid f \,\text{is an SEDF of} G\}$$. 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.