Skip to main content

Research Repository

Advanced Search

Bounding the mim-width of hereditary graph classes

Brettell, N.; Horsfield, J.; Munaro, A.; Paesani, G.; Paulusma, D.

Bounding the mim-width of hereditary graph classes Thumbnail


Authors

N. Brettell

J. Horsfield

A. Munaro

G. Paesani



Contributors

Y. Cao
Editor

M. Pilipczuk
Editor

Abstract

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration. We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied. We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H₁,H₂)-free graphs. We identify several general classes of (H₁,H₂)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H₁,H₂)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H₁ and H₂, the class of (H₁,H₂)-free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H₁,H₂)-free graphs. In particular, we classify the mim-width of (H₁,H₂)-free graphs for all pairs (H₁,H₂) with |V(H₁)| + |V(H₂)| ≤ 8. When H₁ and H₂ are connected graphs, we classify all pairs (H₁,H₂) except for one remaining infinite family and a few isolated cases.

Citation

Brettell, N., Horsfield, J., Munaro, A., Paesani, G., & Paulusma, D. (2020). Bounding the mim-width of hereditary graph classes. In Y. Cao, & M. Pilipczuk (Eds.), 15th International Symposium on Parameterized and Exact Computation (187-199). https://doi.org/10.4230/lipics.ipec.2020.6

Conference Name IPEC 2020
Acceptance Date Sep 30, 2020
Online Publication Date Dec 4, 2020
Publication Date 2020
Deposit Date Jan 16, 2021
Publicly Available Date Mar 29, 2024
Volume 12301
Pages 187-199
Series Title Leibniz International Proceedings in Informatics
Book Title 15th International Symposium on Parameterized and Exact Computation
DOI https://doi.org/10.4230/lipics.ipec.2020.6
Public URL https://durham-repository.worktribe.com/output/1139236

Files





You might also like



Downloadable Citations