Induced Subgraph Isomorphism: Are Some Patterns Substantially Easier Than Others?
Any fixed pattern graph that has a maximum independent set of size k that is disjoint from other maximum independent sets is not easier to detect as an induced subgraph than an independent set of size k. It follows in particular that an induced path on k vertices is not easier to detect than an independent set on ⌈k/2 ⌉ vertices, and that an induced even cycle on k vertices is not easier to detect than an independent set on k/2 vertices. In view of linear time upper bounds on induced paths of length three and four, our lower bound is tight. Similar corollaries hold for the detection of induced complete bipartite graphs and induced complete split graphs.
For an arbitrary pattern graph H on k vertices with no isolated vertices, there is a simple subdivision of H, resulting from splitting each edge into a path of length four and attaching a distinct path of length three at each vertex of degree one, that is not easier to detect or count than an independent set on k vertices, respectively.
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