Describing hereditary properties by forbidden circular orderings

TL;DR

This paper explores characterizations of hereditary graph properties using forbidden circular orderings, demonstrating that many classes can be described finitely and that circular descriptions often offer more natural insights than linear ones.

Contribution

It introduces the concept of characterizing hereditary properties via forbidden circular orderings and compares these with linear orderings, showing many classes can be finitely described in both frameworks.

Findings

Classes like forests, circular-arc graphs, and graphs with circular chromatic number less than k can be characterized by finitely many forbidden circular orderings.

Finitely forbidden circular orderings can be translated into finitely forbidden linear orderings.

Circular order descriptions are often more natural and elegant than linear order descriptions.

Abstract

Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is $k$-colourable if and only if $V(G)$ admits a linear ordering $\le$ with no vertices $v_1 \le \cdots \le v_{k+1}$ such that $v_i v_{i+1} \in E(G)$ for every $i \in \{ 1, \dots, k \}$. In this paper, we study such characterizations when the extra structure is a circular ordering of the vertex set. We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than $k$. In fact, every description by finitely many forbidden circularly ordered graphs can be translated to a description by finitely many forbidden linearly ordered graphs. Nevertheless, our observations underscore the fact that in many cases the circular order descriptions are nicer and more natural.