On signed cycle domination in graphs

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