Results 1 - 10 of 4,088

A Separator Theorem for Planar Graphs

by Richard J. Lipton, Robert E. Tarjan , 1977
"... Let G be any n-vertex 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 461 (1 self) - Add to MetaCart
Let G be any n-vertex 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

On Simultaneous Planar Graph Embeddings

by P. Brass, E. Cenek, C. A. Duncan, A. Efrat, C. Erten, D. Ismailescu, S. G. Kobourov, A. Lubiw, J. S. B. Mitchell - COMPUT. GEOM , 2003
"... We consider the problem of simultaneous embedding of planar graphs. There are two variants ..."
Abstract - Cited by 42 (10 self) - Add to MetaCart
We consider the problem of simultaneous embedding of planar graphs. There are two variants

Planar embedding of planar graphs

by Danny Dolev, Tom Leighton, Howard Trickey - Advances in Computing Research , 1984
"... Planar embedding with minimal area of graphs on an integer grid is an interesting problem in VLSI theory. Valiant [12] gave an algorithm to construct a planar embedding for trees in linear area, he also proved that there are planar graphs that require quadratic area. We fill in a spectrum between Va ..."
Abstract - Cited by 31 (1 self) - Add to MetaCart
Planar embedding with minimal area of graphs on an integer grid is an interesting problem in VLSI theory. Valiant [12] gave an algorithm to construct a planar embedding for trees in linear area, he also proved that there are planar graphs that require quadratic area. We fill in a spectrum between

Planar graphs as VPG-graphs

by Steven Chaplick, Torsten Ueckerdt , 2013
"... A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that th ..."
Abstract - Cited by 5 (3 self) - Add to MetaCart
A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show

Random planar graphs

by Colin McDiarmid, Angelika Steger , Dominic J. A. Welsh - JOURNAL OF COMBINATORIAL THEORY, SERIES B 93 (2005) 187 –205 , 2005
"... We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive i ..."
Abstract - Cited by 66 (20 self) - Add to MetaCart
We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive

Acyclic colorings of planar graphs

by Wayne Goddard - DISCRETE MATH , 1991
"... It is shown that a planar graph can be partitioned into three linear forests. The sharpness of the result is also considered. ..."
Abstract - Cited by 9 (2 self) - Add to MetaCart
It is shown that a planar graph can be partitioned into three linear forests. The sharpness of the result is also considered.

On Random Planar Graphs, the Number of Planar Graphs and Their Triangulations

by Deryk Osthus, Hans Jürgen Prömel, Anusch Taraz, Bender Gao - Jombinatorial Theory, Series B , 2001
"... Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that jPn j n! 75:8 n+o(n) and Bender, Gao and Wormald proved that jPn j n! 26:1 n+o(n) . McDiarmid proved that almost all graphs in Pn have at least 13=7n edges. In this paper, we show that jPn j n! 37 ..."
Abstract - Cited by 28 (1 self) - Add to MetaCart
Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that jPn j n! 75:8 n+o(n) and Bender, Gao and Wormald proved that jPn j n! 26:1 n+o(n) . McDiarmid proved that almost all graphs in Pn have at least 13=7n edges. In this paper, we show that jPn j n

Matched Drawings of Planar Graphs

by Emilio Di Giacomo, Walter Didimo, Marc van Kreveld, Giuseppe Liotta, Bettina Speckmann , 2007
"... A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs all ..."
Abstract - Cited by 7 (2 self) - Add to MetaCart
A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs

Supereulerian planar graphs

by Hong-jian Lai, Mingquan Zhan, Deying Li, Jingzhong Mao , 2003
"... We investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence of spann ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
We investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence

The Random Planar Graph

by Alain Denise, Marcio Vasconcellos, Dominic J. A. Welsh - Congressus Numerantium , 1996
"... We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2-connected and has approximately 2n edges. We present a rs ..."
Abstract - Cited by 38 (1 self) - Add to MetaCart
We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2-connected and has approximately 2n edges. We present a
Results 1 - 10 of 4,088