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

2025-03-10

AI Summary

This paper investigates how hereditary graph properties ℘ can be characterized by finite families ℱ of 2-edge-colored graphs: a graph G belongs to ℘ if and only if it admits a 2-edge-coloring containing no induced 2-edge-colored subgraph isomorphic to any member of ℱ. We establish the first systematic expressive framework for 2-edge-colored graphs and fully classify all hereditary graph classes definable by families ℱ consisting solely of 2-edge-colored graphs on at most three vertices. Our key contributions are: (1) a precise characterization of the correspondence between expressive power and computational complexity; (2) a proof that the membership problem is in P for all families ℱ comprising only 2-edge-colored graphs with ≤2 edges, or those containing a monochromatic path and a colored triangle on three vertices; and (3) the construction of the first NP-complete instance, thereby establishing a tight dichotomy between polynomial-time solvability and NP-completeness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Characterize hereditary properties expressed by 2-edge-coloured graphs.
• Determine polynomial-time solvability of F-free 2-edge-colouring problems.
• Identify NP-complete cases in F-free 2-edge-colouring problems.

Innovation

• Characterizes hereditary properties via 2-edge-coloured graphs.
• Develops polynomial-time algorithms for F-free colouring problems.
• Reduces graph problems to boolean constraint satisfaction.