Either/Or: Using Vertex Cover Structure in Designing FPT-Algorithms - The Case of k-Internal Spanning Tree

To determine if a graph has a spanning tree with few leaves is NP-hard. In this paper we study the parametric dual of this problem, k-INTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time O(2 log k · k + k · n). We also give a 2-approximation algorithm for the problem. However, the main contribution of this paper is that we show the following remarkable structural bindings between k-INTERNAL SPANNING TREE and k-VERTEX COVER: • NO for k-VERTEX COVER implies YES for k-INTERNAL SPANNING TREE. • YES for k-VERTEX COVER implies NO for (2k + 1)-INTERNAL SPANNING TREE. We give a polynomial-time algorithm that produces either a vertex cover of size k or a spanning tree with at least k internal vertices. We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by k-INTERNAL SPANNING TREE. We also briefly discuss the application of this vertex cover methodology to the parametric dual of the DOMINATING SET problem. This design technique seems to apply to many other FPT problems.