New Cases of the Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Paths

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.