Word-Representability of Well-Partitioned Chordal Graphs

TL;DR

This paper studies word-representability in well-partitioned chordal graphs using split decomposition. It proves that every prime component of the minimal split decomposition is a word-representable split graph, enabling a complete characterization and polynomial-time recognition via split-graph theory, with $\mathcal{R}(G)\le 3$ for word-representable instances. It also provides a forbidden-subgraph characterization for circle graphs within this class and identifies which WP chordal graphs have representation number exactly three. By linking split-decomposition structure to word-representability, the work offers efficient tools for recognizing and analyzing this graph class and deepens the connection between split graphs and chordal graphs in the context of word representations.

Abstract

In this paper, we study the word-representability of well-partitioned chordal graphs using split decomposition. We show that every component of the minimal split decomposition of a well-partitioned chordal graph is a split graph. Thus we have a characterization for word-representability of well-partitioned chordal graphs. As a consequence, we prove that the recognition of word-representability of well-partitioned chordal graphs can be done in polynomial time. Moreover, we prove that the representation number of a word-representable well-partitioned chordal graph is at most three. Further, we obtain a minimal forbidden induced subgraph characterization of circle graphs restricted to well-partitioned chordal graphs. Accordingly, we determine the class of word-representable well-partitioned chordal graphs having representation number exactly three.

Figures (1)

• Figure 1: The family of graphs $\mathcal{W}$

Theorems & Definitions (18)

• remark 1
• remark 2
• theorem 1
• proof
• remark 3
• theorem 2: Kitaev_2021Kitaev_2024
• corollary 1
• theorem 3
• proof
• theorem 4