A characterisation of cubic parity graphs

Abstract

A graph is Zm-well-covered if all maximal independent sets have the same cardinality modulo m. Zm-well-covered graphs generalise well-covered graphs, those in which all independent sets have the same cardinality. Z2- well-covered graphs are also called parity graphs. A characterisation of cubic well-covered graphs was given by Campbell, Ellingham and Royle. Here we extend this to a characterisation of cubic Zm-well-covered graphs for all integers m ≥ 2; the most interesting case is m = 2, cubic parity graphs. Our main technique involves minimal non-well-covered graphs, and allows us to build our characterisation as an extension of the existing characterisation of cubic well-covered graphs.