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.
翻译:如果$G不包含$H(不一定是诱导的)子图,则Gog $是不含$H的下方图。 我们着手将具有在捆绑的树枝类和亚紫色图状上的NP完整多米时间可溶解的无H美元下方图分为三个问题分类,但是NP的硬性在边缘分区中得不到保存。 这三个问题是 $k$ 引导的分离路径, $C_ 5$ 聚居和汉密尔顿循环。 虽然我们没有完成分类,但我们显示多米时间和NP的完整界线与其他两个问题不同, $C_ 5$- 聚积。