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 $O(2^{¥#¥mbox{edges}})$ 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).
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 $O(2^{¥#¥mbox{edges}})$ 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).
Original language | English |
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Article number | 10 |
Number of pages | 10 |
Journal | RAIRO - Theoretical Informatics and Applications (RAIRO: ITA) |
Volume | 58 |
Publication status | Accepted/In press - 23 Feb 2024 |
Keywords
- Word-representable graphs
- human-verifiable proofs