On signed cycle domination in graphs

Let G = ( V , E ) be a graph, a function f : E → { − 1 , 1 } is said to be an signed cycle dominating function (SCDF) of G if ∑ e ∈ E ( C ) f ( e ) ≥ 1 holds for any induced cycle C of G . The signed cycle domination number of G is defined as γ s c ′ ( G ) = min { ∑ e ∈ E ( G ) f ( e ) ∣ f is an SCDF of G }. In this paper, we obtain bounds on γ s c ′ ( G ) , characterize all connected graphs G with γ s c ′ ( G ) = | E ( G ) | − 2 , and determine the exact value of γ s c ′ ( G ) for some special classes of graphs G . In addition, we pose some open problems and conjectures.