Simple certifying algorithms for variants of the (2, 1)-colouring problem

Abstract

A (2, 1)-colouring of a graph is a partition of its vertex set into two independent sets and a clique (any of which may be empty). We present forbidden induced subgraph characterizations for some hereditary graph classes obtained from (2, 1)-colouring by adding restrictions between its parts (e.g., there are no edges between the clique and one of the independent sets). The obtained characterizations yield certifying algorithms to recognize these graph classes, which run in time O(|V| + |E|). All the algorithms presented are straightforward to implement using basic data structures.