Results 1  10
of
49,141
On the crossing number of almost planar graphs
 IN PROC. GD ’05, VOLUME 4372 OF LNCS
, 2006
"... Crossing minimization is one of the most challenging algorithmic problems in topological graph theory, with strong ties to graph drawing applications. Despite a long history of intensive research, no practical “good” algorithm for crossing minimization is known (that is hardly surprising, since the ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
the problem itself is NPcomplete). Even more surprising is how little we know about a seemingly simple particular problem: to minimize the number of crossings in an almost planar graph, that is, a graph with an edge whose removal leaves a planar graph. This problem is in turn a building block in an “edge
On the Crossing Number of Almost Planar Graphs
, 2005
"... If G is a plane graph and x, y ∈ V (G), then the dual distance of x and y is equal to the minimum number of crossings of G with a closed curve in the plane joining x and y. Riskin [7] proved that if G0 is a 3connected cubic planar graph, and x, y are its vertices at dual distance d, then the crossin ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
If G is a plane graph and x, y ∈ V (G), then the dual distance of x and y is equal to the minimum number of crossings of G with a closed curve in the plane joining x and y. Riskin [7] proved that if G0 is a 3connected cubic planar graph, and x, y are its vertices at dual distance d
Harmonic Functions On Planar And Almost Planar Graphs And Manifolds, Via Circle Packings
"... . The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If P is a bounded circle packing in R 2 whose contacts graph is a bounded valence triangulation of a disk, then, with probability 1 , ..."
Abstract
 Add to MetaCart
, the simple random walk on P converges to a limit point. Moreover, in this situation any continuous function on the limit set of P extends to a continuous harmonic function on the closure of the contacts graph of P ; that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs
New almostplanar crossingcritical graph families
, 2007
"... We show that, for all choices of integers k> 2 and m, there are simple 3connected kcrossingcritical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of o ..."
Abstract
 Add to MetaCart
of odd degrees at least 5 in crossingcritical graphs remains open. Furthermore, our constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval
New almostplanar crossingcritical graph families
, 2007
"... We show that, for all choices of integers k> 2 and m, there are simple 3connected kcrossingcritical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of o ..."
Abstract
 Add to MetaCart
of odd degrees at least 5 in crossingcritical graphs remains open. Furthermore, our constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval
New infinite families of almostplanar crossingcritical graphs
, 2008
"... We show that, for all choices of integers k> 2 and m, there are simple 3connected kcrossingcritical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of o ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
of odd degrees at least 7 in crossingcritical graphs remains open. Furthermore, our newly constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval [ 3 + 1 5
A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which ..."
Abstract

Cited by 465 (1 self)
 Add to MetaCart
Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which
A fast and high quality multilevel scheme for partitioning irregular graphs
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1998
"... Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. ..."
Abstract

Cited by 1173 (16 self)
 Add to MetaCart
Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc.
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
Abstract

Cited by 498 (68 self)
 Add to MetaCart
the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae
Results 1  10
of
49,141