Graphs with no induced wheel or antiwheel

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. Four families of graphs have repeatedly played important roles in structural graph theory recently. They are called Truemper configurations as they were first used by Truemper in several theorems [9]. These configurations are called pyramids, prisms, thetas and wheels. We will not recall all the definitions, as we do not need all of them here; see Vusković [10] for a very extensive survey on Truemper configurations. It is interesting to know the complexity of deciding whether a graph contains a Truemper configuration of a certain type. The problem is polynomial for pyramids [1]; indeed it is one of the main steps in the polynomial-time recognition algorithm for perfect graphs presented in [1]. On the other hand, the problem is NP-complete for thetas [2] and prisms [7]. Here we will deal only with the fourth Truemper configuration, the wheel. A wheel is a graph that consists of a cycle of length at least 4 plus a vertex that has at least three neighbors on the cycle. Diot, Tavenas and Trotignon [3] proved that it is also NP-complete for wheels, and they mention the question of characterizing the graphs that contain no wheel and no antiwheel but leaves it open. This question is solved here with a complete description of the structure of ∗Partially supported by ANR project STINT under reference ANR-13-BS02-0007.