Characterization of Double-Arborescences and their Minimum-Word-Representants

TL;DR

This paper characterizes double-arborescences as $P_4$-free treelike comparability graphs, explores their structure via split-decomposition trees, and introduces an algorithmic approach to their minimum-word-representants, addressing an open problem in word-representable graphs.

Contribution

It provides a forbidden subgraph characterization of double-arborescences, relates them to $P_4$-free graphs, and offers an algorithmic method for their minimum-word-representants, advancing understanding in word-representable graph classes.

Findings

Double-arborescences are $P_4$-free treelike comparability graphs.

Split-decomposition trees characterize double-arborescences and arborescences.

An algorithmic procedure determines minimum-word-representants for double-arborescences.

Abstract

A double-arborescence is a treelike comparability graph with an all-adjacent vertex. In this paper, we first give a forbidden induced subgraph characterization of double-arborescences, where we prove that double-arborescences are precisely $P_4$-free treelike comparability graphs. Then, we characterize a more general class consisting of $P_4$-free distance-hereditary graphs using split-decomposition trees. Consequently, using split-decomposition trees, we characterize double-arborescences and one of its subclasses, viz., arborescences; a double-arborescence is an arborescence if its all-adjacent vertex is a source or a sink. In the context of word-representable graphs, it is an open problem to find the classes of word-representable graphs whose minimum-word-representants are of length $2n - k$, where $n$ is the number of vertices of the graph and $k$ is its clique number. Contributing to the open problem, we devise an algorithmic procedure and show that the class of double-arborescences is one such class. It seems the class of double-arborescences is the first example satisfying the criteria given in the open problem, for an arbitrary $k$.