Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs

TL;DR

This paper characterizes minimal 2-connected, non-Hamiltonian graphs within split and triangle-free classes, extending previous work on minimal non-Hamiltonian graphs and highlighting the complexity of testing Hamiltonicity.

Contribution

It provides a complete characterization of minimal 2-connected, non-Hamiltonian graphs in split and triangle-free classes, which was previously unknown.

Findings

Characterization of minimal 2-connected, non-Hamiltonian split graphs.

Characterization of minimal 2-connected, non-Hamiltonian triangle-free graphs.

Testing Hamiltonicity remains NP-hard in these classes.

Abstract

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are $2$-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are $2$-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (2) triangle-free graphs. We remark that testing for Hamiltonicity is NP-hard in both of these classes.