An Optimal Algorithm to Detect Sign Compatibility of a Given Sigraph

Abstract

A $\text{signed\,graph}$ (or $\text{sigraph}$ in short) $S$ is a graph $G$ in which each edge $x$ carries a value $s(x) \in \{+1, -1\}$ called its $sign$ denoted specially as $S = (G, s)$. A sigraph $S$ is $\text{sign-compatible}$ if there exists a marking $\mu$ of its vertices such that the end vertices of every negative edge receive ‘-’ signs in $\mu$ and no positive edge in $S$ has both of its ends assigned ‘-’ sign in $\mu$. In this paper, we write algorithms to detect sign-compatibility of a given sigraph and obtain optimal algorithm with complexity $O(n^2)$.