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.
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