Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

TL;DR

This paper characterizes all minimal 2-connected non-Hamiltonian graphs that do not contain wheels, extending previous classifications of HC-obstructions under various graph constraints.

Contribution

It provides a complete characterization of HC-obstructions that are wheel-free, filling a gap in the understanding of minimal non-Hamiltonian graphs with specific induced subgraph restrictions.

Findings

Characterization of all wheel-free HC-obstructions.

Extension of previous classifications to wheel-free graphs.

Provides structural insights into minimal non-Hamiltonian graphs.

Abstract

Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph $H$, a graph $G$ is $H$-free if $G$ has no induced subgraph isomorphic to $H$. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian $\{K_{1,3}, N_{3,1,1}\}$-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs.