Complexity Framework for Forbidden Subgraphs: When Hardness Is Not Preserved under Edge Subdivision
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A graph G is H-subgraph-free if G does not contain H as a (not necessarily induced) subgraph. We make inroads into the classification of three problems for H-subgraph-free graphs that have the properties that they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, yet NP-hardness is not preserved under edge subdivision. The three problems are k-Induced Disjoint Paths, C_5-Colouring and Hamilton Cycle. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs for C_5-Colouring from the other two problems.
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