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

TL;DR

The paper investigates which hereditary graph properties can be expressed via forbidden induced subgraphs in the 2-edge-colouring setting, focusing on three-vertex patterns. It provides complete structural characterizations for two regimes: patterns with at most two edges and patterns including monochromatic $P_3$ with coloured triangles, linking these classes to line graphs of bipartite graphs, incidence graphs, co-bipartite graphs, and clusters. Algorithmically, it establishes polynomial-time recognizability for many $\mathcal{F}$-free colourings through structural reductions and a uniform CSP framework, while identifying NP-complete cases via reductions from constraint problems such as positive 1-in-3 SAT. The work further develops a uniform reduction to boolean CSPs, clarifying when tractable reductions exist (via Schaefer’s theorem) and outlining generalizations to larger patterns and more colours, together with open problems on minimal obstructions and expressivity comparisons with oriented 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.

Figures (7)

• Figure 1: Some celebrities in the world of small graphs.
• Figure 2: The family $\mathcal{J}$ of all $2$-edge-coloured patterns on $3$ vertices; red edges are dashed.
• Figure 3: "The five wonders of the non-elementary world" maffrayJCTB75.
• Figure 4: Connected graphs on 5 vertices containing the diamond but avoiding the remaining celebrities from the world of small graphs (see \ref{['fig:small-graphs']}).
• Figure 5: The landscape of graph classes characterized in Sections \ref{['sec:elementary']} and \ref{['sec:further-elementary']}. In particular, of all graph classes expressible by a set $\mathcal{F}$ containing both monochromatic paths, and $\mathcal{F}\subseteq\{{}^bP_3,{}^rP_3,{}^bK_3,{}^rK_3,{}^{rrb}K_3,{}^{rbb}K_3\}$.

Theorems & Definitions (85)

• Lemma 1
• proof
• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5
• Proposition 6
• Proposition 7