Two edge-disjoint paths with length constraints

Abstract We consider the problem of finding, for two pairs ( s 1 , t 1 ) and ( s 2 , t 2 ) of vertices in an undirected graph, an ( s 1 , t 1 ) -path P 1 and an ( s 2 , t 2 ) -path P 2 such that P 1 and P 2 share no edges and the length of each P i satisfies constraint L i , where L i ∈ { ≤ k i , = k i , ≥ k i , ⁎ } with L i = “ ⁎ ” indicating no length constraint on P i . We regard k 1 and k 2 as parameters and investigate the parameterized complexity of the above problem when at least one of P 1 and P 2 has a length constraint. For the 9 different cases of ( L 1 , L 2 ) , we obtain FPT algorithms for 7 of them by using random partition backed by some structural results. On the other hand, we prove that the problem admits no polynomial kernel for all 9 cases unless N P ⊆ c o N P / p o l y .

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