Computing shortest 12-representants of labeled graphs

TL;DR

This paper introduces an efficient algorithm to find the shortest 12-representant of labeled graphs, improving the understanding of their structure and representation complexity.

Contribution

It presents an $O(n^2)$-time algorithm for computing the shortest 12-representant of a labeled graph, advancing the computational methods for graph representations.

Findings

The algorithm runs in quadratic time.

It guarantees the shortest 12-representant if it exists.

The approach leverages properties of 12-representable graphs.

Abstract

The notion of $12$-representable graphs was introduced as a variant of a well-known class of word-representable graphs. Recently, these graphs were shown to be equivalent to the complements of simple-triangle graphs. This indicates that a $12$-representant of a graph (i.e., a word representing the graph) can be obtained in polynomial time if it exists. However, the $12$-representant is not necessarily optimal (i.e., shortest possible). This paper proposes an $O(n^2)$-time algorithm to generate a shortest $12$-representant of a labeled graph, where $n$ is the number of vertices of the graph.