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

TL;DR

This work characterizes a special class of treelike comparability graphs, double-arborescences, via split-decomposition Trees and forbidden subgraphs, proving they are exactly the $P_4$-free treelike comparability graphs and connecting them to $P_4$-free distance-hereditary graphs through s-leaf-paths. It provides a split-decomposition framework to likewise characterize arborescences and their relation to double-arborescences, including necessary and sufficient conditions on the minimal split-decomposition trees. A central contribution is an algorithmic procedure that constructs minimum-word-representants for double-arborescences, establishing that for a graph on $n$ vertices with clique number $k$, the minimum word length is $2n-k$, thereby addressing an open problem in the theory of word-representable graphs. The results yield both structural characterizations and practical prescriptions for word representations, with implications for enumeration, recognition, and broader understanding of word-representable graph classes.

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 $k$.

Figures (7)

• Figure 1: Examples of treelike comparability graphs in terms of transitive reductions
• Figure 2: A tree $T$
• Figure 3: (a) A graph, and (b) its minimal split-decomposition tree
• Figure 4: Strict-double-arborescence.
• Figure 5: A portion of some $T_{\mathcal{F}}$ with an s-leaf-path $P$ and its accessibility graph $P_4$

Theorems & Definitions (31)

• theorem 1: MR3907778cunningham_2graph-labelled_2012
• theorem 2: cornelsen2009treelike
• theorem 3: MR3907778
• lemma 1: MR3907778
• lemma 2
• proof
• theorem 4
• proof
• lemma 3
• proof