Induced disjoint paths problem in a planar digraph

Abstract

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.