# The Relative Signed Clique Number of Planar Graphs is 8

• Sandip Das
• Soumen Nandi
• Sagnik Sen
• Ritesh Seth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

## Abstract

A simple signed graph $$(G, \varSigma )$$ is a simple graph with a $$+$$ve or a −ve sign assigned to each of its edges where $$\varSigma$$ denotes the set of −ve edges. A cycle is unbalanced if it has an odd number of −ve edges. A vertex subset R of $$(G, \varSigma )$$ 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 $$\omega _{rs}((G, \varSigma ))$$ of $$(G,\varSigma )$$ is the maximum value of |R| where R is a relative signed clique of $$(G,\varSigma )$$. Given a family $$\mathcal {F}$$ of signed graphs, the relative signed clique number is $$\omega _{rs}(\mathcal {F}) = \max \{\omega _{rs}((G,\varSigma ))|(G,\varSigma ) \in \mathcal {F}\}$$. For the family $$\mathcal {P}_3$$ of signed planar graphs, the problem of finding the value of $$\omega _{rs}(\mathcal {P}_3)$$ is an open problem. In this article, we close it by proving $$\omega _{rs}(\mathcal {P}_3)=8$$.

## Keywords

Signed graphs Relative clique number Planar graphs

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Sandip Das
• 1
• Soumen Nandi
• 2
Email author
• Sagnik Sen
• 3
• Ritesh Seth
• 3
1. 1.Indian Statistical InstituteKolkataIndia
2. 2.Birla Institute of Technology and Science Pilani, Hyderabad CampusPilaniIndia
3. 3.Ramakrishna Mission Vivekananda Educational and Research InstituteKolkataIndia