The Relative Signed Clique Number of Planar Graphs is 8

A simple signed graph (G, Sigma) is a simple graph with a +ve or a - ve sign assigned to each of its edges where Sigma denotes the set of -ve edges. A cycle is unbalanced if it has an odd number of - ve edges. A vertex subset R of (G, Sigma) is a relative signed clique if each pair of non-adjacent vertices of R is part of an unbalanced 4-cycle. The relative signed clique number omega(rs)((G, Sigma)) of (G, Sigma) is the maximum value of vertical bar R vertical bar where R is a relative signed clique of (G, Sigma). Given a family F of signed graphs, the relative signed clique number is omega(rs)(F) = max{omega(rs)((G, Sigma))vertical bar(G, Sigma is an element of F}. For the family P-3 of signed planar graphs, the problem of finding the value of omega(rs)(P-3) is an open problem. In this article, we close it by proving omega(rs)(P-3) = 8.