Classification of edge-critical underlying absolute planar cliques for signed graphs

Abstract

International audienceA simple signed graph (G,Σ) is a simple graph G having two different types of edges, positive edges and negative edges, where Σ denotes the set of negative edges of G. A closed walk of a signed graph is positive (resp., negative) if it has even (resp., odd) number of negative edges, taking repeated edges into account. A homomorphism (resp., colored homomorphism) of a simple signed graph to another simple signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks (resp., signs of edges). A simple signed graph (G,Σ) is a signed absolute clique (resp., (0,2)-absolute clique) if any homomorphism (resp., colored homomorphism) of it is an injective function, in which case G is called an underlying signed absolute clique (resp., underlying (0,2)-absolute clique). Moreover, G is edge-critical if G - e is not an underlying signed absolute clique (resp., underlying (0,2)-absolute clique) for any edge e of G. In this article, we characterize all edge-critical outerplanar underlying (0,2)-absolute cliquesand all edge-critical planar underlying signed absolute cliques. We also discuss the motivations and implications of obtaining these exhaustive lists