Human-verifiable proofs in the theory of word-representable graphs

Abstract

A graph is word-representable if it can be represented in a certain way using alternation of letters in words. Word-representable graphs generalise several important and well-studied classes of graphs, and they can be characterised by semi-transitive orientations. Recognising word-representability is an NP-complete problem, and the bottleneck of the theory of word-representable graphs is convincing someone that a graph is non-word-representable, keeping in mind that references to (even publicly available and user-friendly) software are not always welcome. (Word-representability can be justified by providing a semi-transitive orientation as a certificate that can be checked in polynomial time). In the literature, a variety of (usually ad hoc) proofs of non-word-representability for particular graphs, or families of graphs, appear, but for a randomly selected graph, one should expect looking at O(2#edges) orientations and justifying that none of them is semi-transitive. Even if computer would print out all these orientations and would point out what is wrong with each of the orientations, such a proof would be essentially non-checkable by a human. In this paper, we develop methods for an automatic search of human-verifiable proofs of graph non-word-representability. As a proof-of-concept, we provide “short” proofs of non-word-representability, generated automatically by our publicly available user-friendly software, of the Shrikhande graph on 16 vertices and 48 edges (6 “lines” of proof) and the Clebsch graph on 16 vertices and 40 edges (10 “lines” of proof). Producing such short proofs for relatively large graphs would not be possible without the instrumental tool we introduce (allowing to assume orientations of several edges in a graph, not just one edge as it was previously used) that is a game changer in the area. As a bi-product of our studies, we correct two mistakes published multiple times (two graphs out of the 25 non-word-representable graphs on 7 vertices were actually word-representable, while two non-word-representable graphs on 7 vertices were missing).