Tree-layout based graph classes: proper chordal graphs

TL;DR

This work introduces tree-layouts as a natural generalization of vertex layouts to rooted trees and defines graph classes by excluding patterns via ancestor relations. Focusing on proper chordal graphs, the authors characterize them as graphs admitting an indifference tree-layout, and provide a canonical FPQ-hierarchy representation that captures all such layouts rooted at a vertex. They prove polynomial-time recognition for proper chordal graphs and establish a polynomial-time isomorphism test within this class, advancing tractability beyond strongly chordal graphs. The paper also situates proper chordal graphs within the chordal hierarchy, relates them to interval and rooted directed path graphs, and lays out a canonical structural framework (block trees and FPQ-hierarchies) that can drive further algorithmic developments and generalizations of tree-layout based graph classes.

Abstract

Many standard graph classes are known to be characterized by means of layouts (a permutation of its vertices) excluding some patterns. Important such graph classes are among others: proper interval graphs, interval graphs, chordal graphs, permutation graphs, (co-)comparability graphs. For example, a graph $G=(V,E)$ is a proper interval graph if and only if $G$ has a layout $L$ such that for every triple of vertices such that $x\prec_L y\prec_L z$, if $xz\in E$, then $xy\in E$ and $yz\in E$. Such a triple $x$, $y$, $z$ is called an indifference triple and layouts excluding indifference triples are known as indifference layouts. In this paper, we investigate the concept of tree-layouts. A tree-layout $T_G=(T,r,ρ_G)$ of a graph $G=(V,E)$ is a tree $T$ rooted at some node $r$ and equipped with a one-to-one mapping $ρ_G$ between $V$ and the nodes of $T$ such that for every edge $xy\in E$, either $x$ is an ancestor of $y$ or $y$ is an ancestor of $x$. Clearly, layouts are tree-layouts. Excluding a pattern in a tree-layout is defined similarly as excluding a pattern in a layout, but now using the ancestor relation. Unexplored graph classes can be defined by means of tree-layouts excluding some patterns. As a proof of concept, we show that excluding non-indifference triples in tree-layouts yields a natural notion of proper chordal graphs. We characterize proper chordal graphs and position them in the hierarchy of known subclasses of chordal graphs. We also provide a canonical representation of proper chordal graphs that encodes all the indifference tree-layouts rooted at some vertex. Based on this result, we first design a polynomial time recognition algorithm for proper chordal graphs. We then show that the problem of testing isomorphism between two proper chordal graphs is in P, whereas this problem is known to be GI-complete on chordal graphs.

Figures (13)

• Figure 1: The patterns on at most $3$ vertices. $\mathcal{L}(\langle\overline{12},{13},{23}\rangle)$ is the class of chordal graphs.
• Figure 2: The two size $4$ forbidden patterns of cographs.
• Figure 3: A tree-layout $(T,r,\rho)$ of a graph $G=(V,E)$.
• Figure 4: An interval graph $G=(V,E)$ and an interval representation of $G$. The layout $\mathbf{L}_{G}=(P,r,\rho)$ associated to that interval representation is $a \prec_{\mathbf{L}_G} b \prec_{\mathbf{L}_G} x \prec_{\mathbf{L}_G} c \prec_{\mathbf{L}_G} d \prec_{\mathbf{L}_G} e$.
• Figure 5: The $3$-sun graph $G$ on the left, a tree intersection model $\mathbf{M}^{\sf T}_G$ of $G$ in the center and a $\mathcal{P}_{\sf int}$-free tree-layout of $G$ on the right.

Theorems & Definitions (53)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 1
• proof
• Theorem 4
• Lemma 2
• proof