Abstract
The independent set problem is solvable in polynomial time for the graphs not containing the path P k for any fixed k. If the induced path P k is forbidden then the complexity of this problem is unknown for k > 6. We consider the intermediate cases that the induced path P k and some of its spanning supergraphs are forbidden. We prove the solvability of the independent set problem in polynomial time for the following cases: (1) the supergraphs whose minimal degree is less than k/2 are forbidden; (2) the supergraphs whose complementary graph has more than k/2 edges are forbidden; (3) the supergraphs from which we can obtain P k by means of graph intersection are forbidden.
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