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

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

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum