11,519 research outputs found
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
Clique versus Independent Set
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication
complexity asks for the minimum number of cuts separating cliques from stable
sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial
CS-separator, i.e. of size , and addresses the problem of
finding a polynomial CS-separator. This question is still open even for perfect
graphs. We show that a polynomial CS-separator almost surely exists for random
graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a
clique and a stable set) then there exists a constant for which we find a
CS-separator on the class of H-free graphs. This generalizes a
result of Yannakakis on comparability graphs. We also provide a
CS-separator on the class of graphs without induced path of length k and its
complement. Observe that on one side, is of order
resulting from Vapnik-Chervonenkis dimension, and on the other side, is
exponential.
One of the main reason why Yannakakis' CL-IS problem is fascinating is that
it admits equivalent formulations. Our main result in this respect is to show
that a polynomial CS-separator is equivalent to the polynomial
Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition
into k complete bipartite graphs, then its chromatic number is polynomially
bounded in terms of k. We also show that the classical approach to the stubborn
problem (arising in CSP) which consists in covering the set of all solutions by
instances of 2-SAT is again equivalent to the existence of a
polynomial CS-separator
Moduli spaces of metric graphs of genus 1 with marks on vertices
In this paper we study homotopy type of certain moduli spaces of metric
graphs. More precisely, we show that the spaces , which parametrize
the isometry classes of metric graphs of genus 1 with marks on vertices are
homotopy equivalent to the spaces , which are the moduli spaces of
tropical curves of genus 1 with marked points. Our proof proceeds by
providing a sequence of explicit homotopies, with key role played by the
so-called scanning homotopy. We conjecture that our result generalizes to the
case of arbitrary genus.Comment: Topology and its Applications, In Press, Corrected Proof, Available
online 3 August 200
Recovering sparse graphs
We construct a fixed parameter algorithm parameterized by d and k that takes
as an input a graph G' obtained from a d-degenerate graph G by complementing on
at most k arbitrary subsets of the vertex set of G and outputs a graph H such
that G and H agree on all but f(d,k) vertices.
Our work is motivated by the first order model checking in graph classes that
are first order interpretable in classes of sparse graphs. We derive as a
corollary that if G_0 is a graph class with bounded expansion, then the first
order model checking is fixed parameter tractable in the class of all graphs
that can obtained from a graph G from G_0 by complementing on at most k
arbitrary subsets of the vertex set of G; this implies an earlier result that
the first order model checking is fixed parameter tractable in graph classes
interpretable in classes of graphs with bounded maximum degree
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