Ž . The two variable greedoid Tutte polynomial f G; t, z , which was introduced in previous work of the authors, is studied via external activities. Two different partitions of the Boolean lattice of subsets are derived and a feasible set expansion Ž . of f G is developed. All three of these results generalize theorems for matroids. One interval partition yields a characterization of antimatroi...

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 L that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in L. We also prove the following combin...

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 prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every va...

6 The dicycle transversal number τ(D) of a digraph D is the mini7 mum size of a dicycle transversal of D, i. e. a set T ⊆ V (D) such that 8 D − T is acyclic. We study the following problem: Given a digraph D, 9 decide if there is a dicycle B in D and a cycle C in its underlying undi10 rected graph UG(D) such that V (B)∩V (C) = ∅. It is known that there is 11 a polynomial time algorithm for this...

We present the first thorough theoretical analysis of the Transitivity Editing problem on digraphs. Herein, the task is to perform a minimum number of arc insertions or deletions in order to make a given digraph transitive. This problem has recently been identified as important for the detection of hierarchical structure in molecular characteristics of disease. Mixing up Transitivity Editing wi...

Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V (D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x, z) and (y, z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V (G)| + r vertice...

In the ad-hoc radio network model, nodes communicate with their neighbors via signals, without knowing topology of underlying digraph. We study information gathering problem, where each node has a piece called rumor , and objective is to transmit all rumors designated target node. For model any collision detection we provide an O ˜ ( n 1.5 ) deterministic protocol, significantly improving trivi...

A rooted acyclic digraph N with labelled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T . Let Tr(N) denote the set of all trees displayed by the network N . In general, there may be many other networks M such that Tr(M) = Tr(N). A network is regular if it is isomorphic with its cover digraph. This paper shows that if N is...

Using a suitable orientation, we give a short proof of a result of Czumaj and Strothmann [3]: Every 2-edge-connected graph G contains a spanning tree T with the property that dT (v) ≤ dG(v)+3 2 for every vertex v. Trying to find an analogue of this result in the directed case, we prove that every 2-arc-strong digraph D has an out-branching B such that d+B(x) ≤ d+D(x) 2 + 1. As a corollary, ever...

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set asD and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of...