Graph minors. VII. Disjoint paths on a surface
Journal of Combinatorial Theory Series B
Finding $k$ Disjoint Paths in a Directed Planar Graph
SIAM Journal on Computing
Graph minors. XI.: circuits on a surface
Journal of Combinatorial Theory Series B
Induced circuits in planar graphs
Journal of Combinatorial Theory Series B
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Discrete Applied Mathematics - Special issue: Combinatorial Optimization 1992 (CO92)
SWAT '92 Proceedings of the Third Scandinavian Workshop on Algorithm Theory
The equivalence of theorem proving and the interconnection problem
ACM SIGDA Newsletter
Combinatorica
Induced disjoint paths problem in a planar digraph
Discrete Applied Mathematics
The disjoint paths problem in quadratic time
Journal of Combinatorial Theory Series B
Induced disjoint paths in AT-Free graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Induced disjoint paths in claw-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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In this paper, we consider a problem which we call the induced disjoint paths problem (IDPP) for planar graphs. We are given a planar 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. This problem setting is a generalization of the disjoint paths problem, since if we subdivide each edge, then desired disjoint paths would be induced disjoint paths. The problem is motivated by not only the disjoint paths problem but also the recognition of an induced subgraph. The latter has been developed in recent years, and this is actually connected to the Strong Perfect Graph Theorem (Chudnovsky, et al., 2006) [1], and the recognition of the perfect graphs (Chudnovsky, et al., 2005) [2]. Our main result in this paper is to give a linear time algorithm for the IDPP for planar graphs. This generalizes the result by Reed, Robertson, Schrijver and Seymour (1993) [14]. This also gives a polynomial time algorithm to find an induced circuit through given k vertices in planar graphs if one exists when k is fixed. The case k=2 was previously proved by McDiarmid, Reed, Schrijver and Shepherd (1994) [10].