Graphs with no induced wheel or antiwheel

TL;DR

This paper characterizes graphs that lack induced wheels and antiwheels, revealing a simple structure that allows for polynomial-time recognition, contrasting with the NP-complete complexity of detecting wheels alone.

Contribution

It introduces a structural characterization of wheel- and antiwheel-free graphs and provides a polynomial-time recognition algorithm.

Findings

Graphs with no wheels or antiwheels have a simple, recognizable structure.

Recognition of such graphs can be performed in polynomial time.

Detecting induced wheels alone is NP-complete.

Abstract

A wheel is a graph that consists of a chordless cycle of length at least 4 plus a vertex with at least three neighbors on the cycle. It was shown recently that detecting induced wheels is an NP-complete problem. In contrast, it is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.