Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that – every strongly connected digraph D of order n with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph is O(k log k); – it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. All improvements use properties of extremal structures obtained after applying local search and properties of some out-branching decompositions. 1

We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf Out-Tree and Directed Maximum Leaf Spanning Out-Tree in the case of directed graphs. The run time of our algorithm is O(poly(|V |) + 4 k k 2) on undirected graphs, and O(4 k |V |·|E|) on directed graphs. Currently, the fastest algorithms for these problems have run times of O(poly(n) + 6.75 k poly(k)) and 2 O(k log k) poly(n), respectively.

We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arrangement problem for undirected graphs. We state several open problems. 1

Abstract. Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We describe three parameterizations of MinLOB and prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph D of order n and a positive integral parameter k, check whether D contains an outbranching with at most n − k leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k · 16 k) and construct an algorithm of running time O(2 O(k log k) + n 2 log n), which is an ‘additive ’ FPT algorithm. 1

A subgraph T of a digraph D is an out-branching if T is an oriented spanning tree with only one vertex of in-degree zero (called the root). The vertices of T of out-degree zero are leaves. In the Directed k-Leaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an out-branching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for the problem of running time 4 k · n O(1). We describe a new algorithm for the problem of running time 3.72 k · n O(1). In Rooted Directed k-Leaf Problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an out-branching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O(k 3)-size kernel for Rooted Directed k-Leaf. In this paper, we obtain an O(k) kernel for Rooted Directed k-Leaf restricted to acyclic digraphs. 1

The k-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-Leaf-Out-Branching problem. We give the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k-Leaf-Out-Branching where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the k-Leaf-Out-Branching problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for Rooted k-Leaf-Out-Branching immediately imply that the seemingly intractable k-Leaf-Out-Branching problem admits a data reduction to n independent O(k³) kernels. These two results, tractability and intractability side by side, are the first ones separating many-to-one kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.

Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1

A subdigraph T of a digraph D is called an out-tree if T is an oriented tree with just one vertex s of in-degree zero. A spanning outtree is called an out-branching. A vertex x of an out-branching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on out-branchings with minimum and maximum number of leaves. We give short proofs of some well-known theorems. 1

We present an O ( √ opt)-approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence. The result is based upon an O(1)-approximation algorithm for a special class of directed graphs called willows. Incorporating the method for willow graphs as a subroutine in a local improvement algorithm gives the bound for general directed graphs.

The parameterized max-leaf problem on undirected graphs (which is also named the max-leaf spanning-tree problem) is formulated as follows: given an undirected graph G and a parameter k, either construct a spanning tree with at least k leaves for G or report ‘No ’ if such a tree does not exist. The problem also has a version for directed graphs that is named the max-leaf out-branching problem. In this paper, we present a simple branch-and-search algorithm of running time O ∗ (4 k) that solves the max-leaf out-branching problem. This significantly improves the previous best algorithm for the problem that runs in time O ∗ (2 O(k log k)). Our main contributions consist of new observations on the combinatorial structures of the problem and the introduction of a new algorithmic technique that provides new perspectives for design and analysis of parameterized algorithms. Our algorithm of running time O ∗ (4 k) is also applicable to the simpler max-leaf spanning-tree problem, improving the previous best algorithm of running time O ∗ (6.75 k) for the problem. 1