On chordal and perfect plane near-triangulations

TL;DR

This paper characterizes chordal and perfect plane near-triangulations using local structural properties, providing linear time algorithms for their recognition based on induced subgraph conditions.

Contribution

It introduces W-components for decomposing plane near-triangulations and offers new local criteria for chordality and perfectness, enabling efficient recognition algorithms.

Findings

W-near-triangulations are decomposable into W-components in linear time.

Chordality is characterized by the absence of induced wheels of at least five vertices.

Perfectness can be checked via local conditions involving internal vertices and faces.

Abstract

A plane near-triangulation G can be decomposed into a collection of induced subgraphs, described here as the W-components of G, such that G is perfect (respectively, chordal) if and only if each of its W-components is perfect (respectively, chordal). Each W-component is a 2-connected plane near-triangulation, free of edge separators and separating triangles. Graphs satisfying these conditions will be called W-near-triangulations. A linear time decomposition of G into its W-components is achievable using known techniques from the literature. W-near-triangulations have the property that the open neighbourhood of every internal vertex induces a cycle. It follows that a W-near-triangulation H of at least five vertices is non-chordal if and only if it contains an internal vertex. This yields a local structural characterization that a plane near-triangulation G is chordal if and only if it does not contain an induced wheel of at least five vertices. For W-near-triangulations that are free of induced wheels of five vertices, we derive a similar local criteria, that depends only on the neighbourhoods of individual vertices and faces, for checking perfectness. We show that a W-near-triangulation H that is free of any induced wheel of five vertices is perfect if and only if there exists neither an internal vertex x, nor a face f such that, the neighbours of x or f induces an odd hole. The above characterization leads to a linear time algorithm for determining perfectness of this class of graphs.