Characterization and linear-time detection of minimal obstructions to concave-round graphs and the circular-ones property

TL;DR

This paper characterizes concave-round graphs through minimal forbidden subgraphs, provides a linear-time detection algorithm, and explores related properties like the circular-ones property, advancing understanding of circular-arc graph classes.

Contribution

It offers the first minimal forbidden subgraph characterization for concave-round graphs and linear-time algorithms for detecting such obstructions.

Findings

Minimal forbidden induced subgraph characterization for concave-round graphs

Linear-time detection of forbidden subgraphs in non-concave-round graphs

Characterizations of the circular-ones property via minimal forbidden submatrices

Abstract

A graph is concave-round if its vertices can be circularly enumerated so that the closed neighbourhood of each vertex is an interval in the enumeration. In this work, we give a minimal forbidden induced subgraph characterization for the class of concave-round graphs, solving a problem posed by Bang-Jensen, Huang, and Yeo [SIAM J Discrete Math, 13:179--193, 2000]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any given graph that is not concave-round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices for the circular-ones property for rows and for the circular-ones property for rows and columns and show that, also for both variants of the property, one of the corresponding forbidden submatrices can be found (if present) in any given matrix in linear time. We make some final remarks regarding connections to some classes of circular-arc graphs.