Full homomorphisms to graph classes

Abstract

Given a family of graphs $\mathcal{F}$, we define a graph $G$ to be fully $\mathcal{F}$-colourable if $G$ admits a full homomorphism to some $F$ in $\mathcal{F}$. We approach the problem of determining when a graph is fully $\mathcal{F}$-colourable in terms of minimal forbidden induced subgraphs. We provide general results which allow to obtain the exact families of forbidden induced subgraphs for full $\mathcal{F}$-colouring when $\mathcal{F}$ is among some well-known families, such as threshold, trivially perfect, split, chordal, interval and strongly chordal graphs, as well as forests. Traditionally, these questions have been studied for a single graph $H$, not a family. Motivated by our results on the family of forests, we contribute to this research by focusing on the case of a single centipede.