On The Signed Cycle Domination Problem Of Planar Graphs

Let G= (V, E) be a graph. A function f:E→{-1,+1} is called the signed cycle domination function (SCDF) of G if f(C)=∑e∈E(C) f(e)≥1 for every induced cycle of G. The signed induced domination number of G is defined asγ'SC(G)=min{f(G)|f is an SCDF of G}.In this paper, we study the signed cycle number on maximum planar graphs,2-con-nected planar graphs and graphs withδ(G)= 3. The main results as follows:(1) Let G (|V|= n) be a maximum planar graph,γ'SC(G)= n-2 if and only if every induced cycle of G is C3.(2) Let G (|V|= n) be a maximum planar graph,γ'SC(G)≥n if and only if G has an induced cycle Ck (k≥4).(3) If G is a 2-connected planar graph, thenγ'SC(G)≥1.(4) Let G be a graph withδ(G)=3, ifγ'SC(G) has a constant bound, thenγ'SC(G)≥2.The above results on maximum planar graphs negative a conjecture in B. Xu [On signed cycle domination in graphs, Discrete Math.309(2009) 1007-1012].