On the expressive power of $2$-edge-colourings of graphs

Abstract

Given a finite set of $2$-edge-coloured graphs $\mathcal F$ and a hereditary property of graphs $\mathcal{P}$, we say that $\mathcal F$ expresses $\mathcal{P}$ if a graph $G$ has the property $\mathcal{P}$ if and only if it admits a $2$-edge-colouring not having any graph in $\mathcal F$ as an induced $2$-edge-coloured subgraph. We show that certain classic hereditary classes are expressible by some set of $2$-edge-coloured graphs on three vertices. We then initiate a systematic study of the following problem. Given a finite set of $2$-edge-coloured graphs $\mathcal F$, structurally characterize the hereditary property expressed by $\mathcal F$. In our main results we describe all hereditary properties expressed by $\mathcal F$ when $\mathcal F$ consists of 2-edge-coloured graphs on three vertices and (1) patterns have at most two edges, or (2) $\mathcal F$ consists of both monochromatic paths and a set of coloured triangles. On the algorithmic side, we consider the $\mathcal F$-free colouring problem, i.e., deciding if an input graph admits an $\mathcal F$-free $2$-edge-colouring. It follows from our structural characterizations, that for all sets considered in (1) and (2) the $\mathcal F$-free colouring problem is solvable in polynomial time. We complement these tractability results with a uniform reduction to boolean constraint satisfaction problems which yield polynomial-time algorithms that recognize most graph classes expressible by a set $\mathcal F$ of $2$-edge-coloured graphs on at most three vertices. Finally, we exhibit some sets $\mathcal F$ such that the $\mathcal F$-free colouring problem is NP-complete.