107 research outputs found
The Number of Seymour Vertices in Random Tournaments and Digraphs
Seymour's distance two conjecture states that in any digraph there exists a
vertex (a "Seymour vertex") that has at least as many neighbors at distance two
as it does at distance one. We explore the validity of probabilistic statements
along lines suggested by Seymour's conjecture, proving that almost surely there
are a "large" number of Seymour vertices in random tournaments and "even more"
in general random digraphs.Comment: 14 page
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple dicycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but have a vertex
of minimum outdegree that has the Second Neighborhood Property.Comment: fixed an error in an earlier version and made structural change
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple cycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but that local
simplicity can, in some cases, also imply the existence of a Seymour vertex in
the original digraph.Comment: This is the version accepted for publication in Opuscula Mathematic
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 pathwidth of almost semicomplete digraphs
We call a digraph {\em -semicomplete} if each vertex of the digraph has at
most non-neighbors, where a non-neighbor of a vertex is a vertex such that there is no edge between and in either direction.
This notion generalizes that of semicomplete digraphs which are
-semicomplete and tournaments which are semicomplete and have no
anti-parallel pairs of edges. Our results in this paper are as follows. (1) We
give an algorithm which, given an -semicomplete digraph on vertices
and a positive integer , in time either
constructs a path-decomposition of of width at most or concludes
correctly that the pathwidth of is larger than . (2) We show that there
is a function such that every -semicomplete digraph of pathwidth
at least has a semicomplete subgraph of pathwidth at least .
One consequence of these results is that the problem of deciding if a fixed
digraph is topologically contained in a given -semicomplete digraph
admits a polynomial-time algorithm for fixed .Comment: 33pages, a shorter version to appear in ESA 201
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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