Recognizing Mim-Width and Its Variants Is NP-Complete at Every Fixed Width
Year: 2026
Type: preprint
Abstract: This preprint investigates the computational complexity of recognizing graphs of bounded mim-width and three related graph-width parameters: sim-width, one-sided mim-width (omim-width), and Omim-width. The manuscript presents NP-completeness results for deciding whether each of these parameters is at most any fixed integer k ≥ 1. The central contribution is a proof of NP-completeness for recognizing graphs of mim-width at most one, addressing a recognition problem left open by previous work. The reduction proceeds from NAE-3SAT through a closed-hierarchy reconstruction problem on ordered graphs, using a rigid-frame construction and a hub-augmented exterior chain. A second contribution establishes polynomial-time width-shift transformations that extend width-one recognition hardness to every fixed positive width for all four parameters. These constructions build on earlier width-shift techniques while addressing parameters whose cut values also depend on edges within the sides of a cut. The paper additionally examines structural limitations of local gap-amplification constructions and identifies directions for further research. Version 3 refines the discussion of closely related literature, clarifies the positioning of the contributions, and updates the accompanying supplementary materials. The supplementary archive provides computational verification code, supporting certificates, and experimental logs. These checks complement the mathematical arguments but do not replace independent proof verification. This manuscript is a preprint and has not undergone peer review.
Author: Kadri, Haitam
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225414299