Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves

Improved self-reduction algorithms for graphs with bounded treewidth

AbstractRecent results of Robertson and Seymour show that every class that is closed under taking of minors can be recognized in O(n3) time. If there is a fixed upper bound on the treewidth of the graphs in the class, i.e., if there is a planar graph not in the class, then the class can be recognized in O(n2) time. However, this result is nonconstructive in two ways: the algorithm only decides on membership, but does not construct “a solution”, e.g., a linear ordering, decomposition or embedding; and no method is given to find the algorithms. In many cases, both nonconstructive elements can be avoided, using techniques of Brown (1989) and Fellows and Langston (1989), based on self-reduction. In this paper we introduce two techniques that help to reduce the running time of self-reduction algorithms. With the help of these techniques we show that there exist O(n2) algorithms that decide on membership and construct solutions for treewidth, pathwidth, search number, vertex search number, node search number, cutwidth, modified cutwidth, vertex separation number, gate matrix layout, and progressive black–white pebbling, where in each case the parameter k is a fixed constant

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2