48 research outputs found
On the existence and number of -kings in -quasi-transitive digraphs
Let be a digraph and an integer. We say that
is -quasi-transitive if for every directed path in
, then or . Clearly, a
2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense.
Bang-Jensen and Gutin proved that a quasi-transitive digraph has a 3-king
if and only if has a unique initial strong component and, if has a
3-king and the unique initial strong component of has at least three
vertices, then has at least three 3-kings. In this paper we prove the
following generalization: A -quasi-transitive digraph has a -king
if and only if has a unique initial strong component, and if has a
-king then, either all the vertices of the unique initial strong
components are -kings or the number of -kings in is at least
.Comment: 17 page
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least 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
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . 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 of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
On the Small Quasi-kernel conjecture
The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph
the existence of a quasi-kernel, i.e., an independent -out-dominating
vertex set. In the same spirit, the Small Quasi-kernel Conjecture, proposed by
Erd\H{o}s and Sz\'ekely in 1976, asserts the existence of a quasi-kernel of
order at most if does not have sources. Despite repeated
efforts, the conjecture remains wide open.
This work contains a number of new results towards the conjecture. In our
main contribution we resolve the conjecture for all directed graphs without
sources containing a kernel in the second out-neighborhood of a quasi-kernel.
Furthermore, we provide a novel strongly connected example demonstrating the
asymptotic sharpness of the conjecture. Additionally, we resolve the conjecture
in a strong form for all directed unicyclic graphs.Comment: 12 pages, 1 figur