1,049 research outputs found
Clique Cover Width and Clique Sum
For a clique cover in the undirected graph , the clique cover graph of
is the graph obtained by contracting the vertices of each clique in
into a single vertex. The clique cover width of G, denoted by , is the
minimum value of the bandwidth of all clique cover graphs of . When is
the clique sum of and , we prove that
New separation theorems and sub-exponential time algorithms for packing and piercing of fat objects
For a collection of objects in , let the packing and
piercing numbers of , denoted by , and , respectively, be the largest number of pairwise disjoint objects in
, and the smallest number of points in that are common to all
elements of , respectively. When elements of are fat objects
of arbitrary sizes, we derive sub-exponential time algorithms for the NP-hard
problems of computing and , respectively,
that run in and
time, respectively, and storage. Our main tool which is interesting in its own way, is a new
separation theorem. The algorithms readily give rise to polynomial time
approximation schemes (PTAS) that run in
time and storage. The results favorably compare with many related
best known results. Specifically, our separation theorem significantly improves
the splitting ratio of the previous result of Chan, whereas, the
sub-exponential time algorithms significantly improve upon the running times of
very recent algorithms of Fox and Pach for packing of spheres.Comment: 28th European Workshop on Computational Geometry,2012 - Assisi,
Perugia, Italy, 269-27
New representation results for planar graphs
A universal representation theorem is derived that shows any graph is the
intersection graph of one chordal graph, a number of co-bipartite graphs, and
one unit interval graph. Central to the the result is the notion of the clique
cover width which is a generalization of the bandwidth parameter. Specifically,
we show that any planar graph is the intersection graph of one chordal graph,
four co-bipartite graphs, and one unit interval graph. Equivalently, any planar
graph is the intersection graph of a chordal graph and a graph that has {clique
cover width} of at most seven. We further describe the extensions of the
results to graphs drawn on surfaces and graphs excluding a minor of crossing
number of at most one.Comment: 29th European Workshop on Computational Geometry March 17-20, 2013,
177-18
Unit Incomparability Dimension and Clique Cover Width in Graphs
For a clique cover in the undirected graph , the {\it clique cover
graph} of is the graph obtained by contracting the vertices of each clique
in into a single vertex. The {\it clique cover width} of , denoted by
, is the minimum value of the bandwidth of all clique cover graphs in
. Any with is known to be an incomparability graph, and hence
is called, a {\it unit incomparability graph}. We introduced the {\it unit
incomparability dimension of }, denoted by, to be the smallest
integer so that there are unit incomparability graphs with
, so that . We prove a
decomposition theorem establishing the inequality .
Specifically, given any , there are unit incomparability graphs
with so that and . In addition, is co-bipartite, for .
Furthermore, we observe that , where is the number
of leaves in a largest induced star of , and use Ramsey Theory to give an
upper bound on , when is represented as an intersection graph using
our decomposition theorem. Finally, when is an incomparability graph we
prove that
Bounds for the Clique Cover Width of Factors of the Apex Graph of the Planar Grid
The {\it clique cover width} of , denoted by , is the minimum
value of the bandwidth of all graphs that are obtained by contracting the
cliques in a clique cover of into a single vertex. For let
be a graph with , and let be a graph with and
, then we write and call each
a factor of . We are interested in the case where is
chordal, and for each factor is "small". Here we
show a negative result. Specifically, let be the graph obtained
by joining a set of apex vertices of degree to all vertices of an
grid, and then adding some possible edges among these vertices.
We prove that if , with being chordal,
then, , where is a constant. Furthermore, for , we construct a
chordal graph and a graph with so that
. Finally, let be the clique sum graph of
, where the underlying grid is
and the sum is taken at apex vertices. Then, we show ,
where, is chordal and . The
implications and applications of the results are discussed, including
addressing a recent question of David Wood
Largest reduced neighborhood clique cover number revisited
Let be a graph and . The largest reduced neighborhood clique
cover number of , denoted by , is the largest, overall
-shallow minors of , of the smallest number of cliques that can cover
any closed neighborhood of a vertex in . It is known that
, where is an incomparability graph and is
the number of leaves in a largest shallow minor which is isomorphic to an
induced star on leaves. In this paper we give an overview of the
properties of including the connections to the greatest
reduced average density of , or , introduce the class
of graphs with bounded neighborhood clique cover number, and derive a simple
lower and an upper bound for this important graph parameter. We announce two
conjectures, one for the value of , and another for a
separator theorem (with respect to a certain measure) for an interesting class
of graphs, namely the class of incomparability graphs which we suspect to have
a polynomial bounded neighborhood clique cover number, when the size of a
largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern
Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic
University, Boca Raton, March 201
A new upper bound for the clique cover number with applications
Let and , denote the size of a largest independent set
and the clique cover number of an undirected graph . Let be an interval
graph with and , and let denote the
maximum of overall induced subgraphs
of that are cliques in . The main result of this paper is to prove that
for any graph
where, is the size of a largest independent set in . We further
provide a generalization that significantly unifies or improves some past
algorithmic and structural results concerning the clique cover number for some
well known intersection graphs
A new separation theorem with geometric applications
Let be an undirected graph with a measure function
assigning non-negative values to subgraphs so that does not exceed
the clique cover number of . When satisfies some additional natural
conditions, we study the problem of separating into two subgraphs, each
with a measure of at most by removing a set of vertices that can be
covered with a small number of cliques . When ,
where is a graph with , and is a chordal graph with , we prove that there is a
separator that can be covered with cliques in ,
where is a parameter similar to the bandwidth, which arises from
the linear orderings of cliques covers in . The results and the methods
are then used to obtain exact and approximate algorithms which significantly
improve some of the past results for several well known NP-hard geometric
problems. In addition, the methods involve introducing new concepts and hence
may be of an independent interest.Comment: Proceedings of EuroCG 2010, Dortmund, Germany, March 22-24, 201
On the largest reduced neighborhood clique cover number of a graph
Let be a graph and . A new graph parameter termed the largest
reduced neighborhood clique cover number of , denoted by ,
is introduced. Specifically, is the largest, overall
-shallow minors of , of the smallest number of cliques that can cover
any closed neighborhood of a vertex in . We verify that
when is chordal, and, , where is an
incomparability graph that does not have a shallow minor which is
isomorphic to an induced star on leaves. Moreover, general properties of
including the connections to the greatest reduced average
density of , or are studied and investigated. For
instance we show where is the size of a largest complete graph which is
a of . Additionally we prove that largest ratio of any minimum
clique cover to the maximum independent set taken overall minors of is
a lower bound for . We further introduce the class of bounded
neighborhood clique cover number for which has a finite
value for each and verify the membership of geometric intersection
graphs of fat objects (with no restrictions on the depth) to this class. The
results support the conjecture that the class graphs with polynomial bounded
neighborhood clique cover number may have separator theorems with respect to
certain measures.Comment: A portion of these results were presented at the 47th Southeast
Conference on Combinatorics, Graph Theory and Computing, March 7-11, 2016 and
will appear in the conference proceedings, Congressus Numerantium (2016
On the strict endoscopic part of modular Siegel threefolds
In this paper we study the non-holomorphic strict endoscopic parts of inner
cohomology spaces of a modular Siegel threefold respect to local systems. First
we show that there is a non-zero subspace of the strict endoscopic part such
that it is constructed by global theta lift of automorphic froms of
. Secondly, we present an explicit analytic
calculation of levels of lifted forms into GSp(4), based on the paramodular
representations theory for . Finally, we prove the conjecture, by C.
Faber and G. van der Geer, that gives a description of the strict endoscopic
part for Betti cohomology and (real) Hodge structures in the category of mixed
Hodge structures, in which the modular Siegel threefold has level structure
one
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