State Canonization and Early Pruning in Width-Based Automated Theorem Proving

2026-05-10

AI Summary

This work proposes an automated framework based on tree decomposition for efficiently verifying graph-theoretic conjectures, particularly on graph classes with bounded treewidth or pathwidth. By integrating dynamic programming, state canonization, and an early pruning strategy tailored to subgraph-closed properties and their implication relations, the method substantially reduces the search space. As a demonstration of its effectiveness, the framework automatically verifies Reed’s conjecture for triangle-free graphs of pathwidth at most 5 and treewidth at most 3—the first such automated confirmation—and further produces counterexamples to certain strengthened but invalid variants of the conjecture. These results highlight the framework’s practical utility and novelty in advancing automated exploration and validation of graph-theoretic conjectures.

Abstract

Width-based automated theorem proving is a framework where counterexamples to graph-theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time \(2^{2^{k^{O(1)}}}\) on the class of graphs of treewidth at most \(k\). In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counterexample of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form \(\mathcal{P}_1 \rightarrow \mathcal{P}_2\), for graph properties \(\mathcal{P}_1\) and \(\mathcal{P}_2\), where \(\mathcal{P}_1\) is a property closed under subgraphs. As a concrete application, we use our framework in the study of graph-theoretic conjectures related to coloring triangle-free graphs. In particular, our algorithm is able to show that Reed's conjecture for triangle-free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counterexamples to invalid strengthenings of Reed's conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.

Problem

• automated theorem proving
• graph-theoretic conjectures
• treewidth
• state space explosion
• Reed's conjecture

Innovation

• state canonization
• early pruning
• width-based automated theorem proving
• tree decomposition
• graph-theoretic conjectures