We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.

Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

Dealing with the NP-complete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set on planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. This answers an open question from previous work on the parameterized complexity of Dominating Set on planar graphs.

We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem.

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.

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Many NP-complete problems on planar graphs are “fixed-parameter tractable:” Recent theoretical work provided tree decomposition based fixed-parameter algorithms exactly solving various parameterized problems on planar graphs, among others Vertex Cover, in time O(c √ k n). Here, c is some constant depending on the graph problem to be solved, n is the number of graph vertices, and k is the problem parameter (for Vertex Cover this is the size of the vertex cover). In this paper, we present an experimental study for such tree decomposition based algorithms focusing on Vertex Cover. We demonstrate that the tree decomposition based approach provides a valuable way of exactly solving Vertex Cover on planar graphs. Doing so, we also demonstrate the impressive power of the so-called Nemhauser/Trotter theorem which provides a Vertex Cover-specific, extremely useful data reduction through polynomial time preprocessing. Altogether, this underpins the practical importance of the underlying theory. 1

Abstract. 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 4k log k · k 7/2 + k 2 · n 2). 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.

Given a graph G and an integer k ≥ 0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.