Word-Representable Graphs and Locality of Words

TL;DR

This paper explores the relationship between $k$-representable graphs and $k$-local words, establishing new bounds and properties, including hereditary characteristics, speed, and clique-width of these graph classes.

Contribution

It proves that graphs representable by $k$-local words are $(k+1)$-representable and analyzes their hereditary properties, speed, and clique-width.

Findings

Graphs representable by $k$-local words are $(k+1)$-representable.

Graphs in these classes have bounded clique-width.

These classes belong to the factorial layer in complexity.

Abstract

In this work, we investigate the relationship between $k$-repre\-sentable graphs and graphs representable by $k$-local words. In particular, we show that every graph representable by a $k$-local word is $(k+1)$-representable. A previous result about graphs represented by $1$-local words is revisited with new insights. Moreover, we investigate both classes of graphs w.r.t. hereditary and in particular the speed as a measure. We prove that the latter ones belong to the factorial layer and that the graphs in this classes have bounded clique-width.