Signed Cycle Domination in Planar Graphs

Abstract

Let $G=(V,E)$ be a graph. A function $f:E\rightarrow \{-1,1\}$ is called a signed cycle dominating function (SCDF) if $\sum \limits _{e\in E(C)} f(e)\ge 1$ for every induced cycle C in G. The signed cycle domination number $\sigma (G)$ is defined as $\sigma (G)=\min \left\{ \sum \limits _{e\in E}f(e): f\right.$ is an SCDF of $G\Big \}.$ In this paper, we prove that for any positive integer $\ell$ with $n-2 \le \ell \le 2n-6$, there exists a maximal planar graph G of order n such that $\sigma (G)=\ell$. We also prove that the problem of determining the signed cycle domination number is NP-complete.