On the complexity of testing for odd holes and induced odd paths
Discrete Mathematics
Finding $k$ Disjoint Paths in a Directed Planar Graph
SIAM Journal on Computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Polynomial Solution to the Undirected Two Paths Problem
Journal of the ACM (JACM)
The equivalence of theorem proving and the interconnection problem
ACM SIGDA Newsletter
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The induced disjoint paths problem
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Finding multiple induced disjoint paths in general graphs
Information Processing Letters
A linear time algorithm for the induced disjoint paths problem in planar graphs
Journal of Computer and System Sciences
Finding an induced subdivision of a digraph
Theoretical Computer Science
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As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s"1,t"1),...,(s"k,t"k)}. The objective is to find k paths P"1,...,P"k such that P"i is a path from s"i to t"i and P"i and P"j have neither common vertices nor adjacent vertices for any distinct i,j. The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph. As an application of our first result, we show that we can find in polynomial time certain structures called a ''hole'' and a ''theta'' in a planar graph.