Complexity Classification of Forbidden Subgraph Problems: Edge Subdivision and the "H"-Graphs

核心概念

The paper studies the complexity of graph problems that are solvable in polynomial time on graph classes of bounded treewidth and NP-complete on subcubic graphs, but whose NP-hardness is not preserved under edge subdivision. It provides a rich complexity landscape for such problems on H-subgraph-free graph classes.

摘要

The paper investigates graph problems that satisfy the first two conditions of the complexity framework proposed in prior work (efficient on bounded treewidth, NP-complete on subcubic graphs), but fail to satisfy the third condition (NP-hardness preserved under edge subdivision).
The authors focus on four such problems: k-Induced Disjoint Paths, C5-Colouring, Hamilton Cycle, and Star 3-Colouring. They show that the boundary between polynomial time and NP-complete differs among these problems and differs from problems that satisfy all three conditions of the framework.
For k-Induced Disjoint Paths, the problem is in P on H1-subgraph-free and H2-subgraph-free graphs, but NP-complete on (H4, ..., Hℓ-1)-subgraph-free graphs for ℓ > 4.
For C5-Colouring, the problem is in P on H3-subgraph-free graphs, but NP-complete for (Hi: i = 1 or 2 mod 3)-subgraph-free graphs.
Hamilton Cycle is in P for the class of H1-subgraph-free graphs.
For Star 3-Colouring, the problem is in P for (H1, H2, H3)-subgraph-free graphs, but NP-complete for (Hi: i is odd)-subgraph-free graphs. Additionally, it is in P for (Hi: i is even)-subgraph-free graphs.
The authors also provide dichotomy results for C5-Colouring and Star 3-Colouring based on the Hi graphs modulo 3 and 2, respectively.

統計資料

None.

引述

None.

從以下內容提煉的關鍵洞見

Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-graphs

by Vadim Lozin,... 於 arxiv.org 05-07-2024

https://arxiv.org/pdf/2211.14214.pdf

深入探究

What is the complexity of k-Induced Disjoint Paths on H3-subgraph-free graphs?

In the context provided, the complexity of k-Induced Disjoint Paths on H3-subgraph-free graphs is still an open problem. The resolution of this question would provide a dichotomy for k-Induced Disjoint Paths on Hi-subgraph-free graphs. The study delves into the classification of the complexity landscape among problems for H-subgraph-free graph classes, focusing on k-Induced Disjoint Paths as one of the key problems. The goal is to determine whether this problem can be efficiently solved on H3-subgraph-free graphs or if it remains NP-complete.

What is the complexity of C5-Colouring on Hi-subgraph-free graphs, when i = 0 mod 3?

The complexity of C5-Colouring on Hi-subgraph-free graphs, specifically when i = 0 mod 3, is a significant area of interest within the research framework discussed. The study aims to understand the computational complexity of C5-Colouring when certain conditions are met, such as the value of i being congruent to 0 modulo 3. By exploring this specific scenario, researchers seek to determine whether C5-Colouring can be efficiently solved on Hi-subgraph-free graphs or if it remains NP-complete under these conditions.

What is the complexity of Star 3-Colouring on Hi-subgraph-free graphs, when i = 0 mod 2?

The complexity of Star 3-Colouring on Hi-subgraph-free graphs, particularly when i = 0 mod 2, is another key aspect of the research discussed in the context. By focusing on this specific condition where i follows a particular modulo pattern, researchers aim to investigate whether Star 3-Colouring exhibits different computational complexities on Hi-subgraph-free graphs. The study delves into understanding whether Star 3-Colouring can be efficiently solved or if it remains NP-complete under the specified circumstances.