Detecting wheels

Abstract

A wheel is a graph made of a cycle of length at least 4 together with a vertex that has at least three neighbors in the cycle. We prove that the problem whose instance is a graph G and whose question is "does G contains a wheel as an induced subgraph" is NP-complete. We also settle the complexity of several similar problems.

Wheel-free planar graphs

“…Since u 3 is not the center of claw, u 1 z is an edge and thus u 1 is a corner, a contradiction to (3). This proves (5). Since G does not admit a clique cutset, it follows that u is non-adjacent to at least one of x, y.…”

“…Also, the four classes have polynomial time recognition algorithms, so one could conjecture that so does the class of wheel-free graphs. But it is proved in [5] that it is NP-hard to recognize them. All this suggest that possibly, no structural description of wheel-free graphs exists.…”

The structure of (theta, pyramid, 1‐wheel, 3‐wheel)‐free graphs

“…Maffray and Trotignon show that detecting whether a graph contains a prism is NP‐complete. Also, detecting whether a graph contains a wheel is NP‐complete, as shown by Diot et al . In fact, they prove that the problem remains NP‐complete even when restricted to bipartite graphs.…”

“…In , Maffray et al give an ‐time algorithm that decides whether a graph contains a theta or a pyramid. Recall that deciding whether a graph contains a 1‐wheel is NP‐complete . In Lemma , we give an ‐time algorithm that decides whether a graph contains a theta, a pyramid, or a 1‐wheel.…”

“…In [15], Maffray et al give an n ( ) 7 -time algorithm that decides whether a graph contains a theta or a pyramid. Recall that deciding whether a graph contains a 1-wheel is NP-complete [7].…”