On the Parameterized Complexity for Token Jumping on Graphs
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Suppose that we are given two independent sets I 0 and I r of a graph such that ∣ I 0 ∣ = ∣ I r ∣, and imagine that a token is placed on each vertex in I 0. Then, the token jumping problem is to determine whether there exists a sequence of independent sets which transforms I 0 into I r so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I 0 and I r with the minimum number of token movements.
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