Hamiltonicity Parameterized by Mim-Width is (Indeed) Para-NP-Hard

TL;DR

This paper establishes that solving Hamiltonian Path and Cycle problems remains NP-hard even on graphs with bounded mim-width 26, resolving a previous gap in the literature regarding their complexity.

Contribution

It proves the para-NP-hardness of Hamiltonicity problems on graphs with mim-width 26, correcting a previous incomplete proof.

Findings

NP-hardness of Hamiltonian problems on graphs with mim-width 26

Provides a complete proof filling previous gaps

Shows hardness persists even with a given linear mim-width order

Abstract

We prove that Hamiltonian Path and Hamiltonian Cycle are NP-hard on graphs of linear mim-width 26, even when a linear order of the input graph with mim-width 26 is provided together with input. This fills a gap left by a broken proof of the para-NP-hardness of Hamiltonicity problems parameterized by mim-width.