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A LeftFirst Search Algorithm for Planar Graphs
 Discrete Computational Geometry
, 1995
"... We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we o ..."
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Cited by 20 (4 self)
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We give an O(jV (G)j) time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graph G so that (i) no two segments have an interior point in common, (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovic. The edges of any maximal bipartite plane graph G with outer face bwb w can be colored by two colors such that the color classes form spanning trees of G b and G b , respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear time algorithm for constructing bipolar orientations of 2connected plane graphs.
Characterizations of arboricity of graphs
 Ars Combinatorica
"... The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spannin ..."
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Cited by 12 (0 self)
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The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of trees or forests dates back to NashWilliams and Tutte [6, 7, 11]. Since then, many authors have examined various tree decompositions of classes of graphs (for example [2, 8]). The aim of this paper is to give several characterizations for
Binary labelings for plane quadrangulations and their relatives
, 2007
"... Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for tr ..."
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Cited by 11 (7 self)
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Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2book. Furthermore, as Schnyder labelings have been extended to 3connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2connected bipartite graphs. AMS subject classification: 05C78
Partitions of graphs into trees
 IN PROCEEDINGS OF GRAPH DRAWING’06 (KARLSRUHE), VOLUME 4372 OF LNCS
, 2007
"... In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) ..."
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Cited by 3 (0 self)
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In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3tree partition of the inner edges. Maximal planar bipartite graphs have a 2tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NPhardness of the ktree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].
A binary labeling for plane Laman . . .
, 2008
"... We present binary labelings for the angles of quadrangulations and plane Laman graphs, which are in analogy with Schnyder labelings for triangulations [W. Schnyder, Proc. 1st ACMSIAM Symposium on Discrete Algorithms, 1990] and imply a special tree decomposition for quadrangulations. In particular, ..."
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We present binary labelings for the angles of quadrangulations and plane Laman graphs, which are in analogy with Schnyder labelings for triangulations [W. Schnyder, Proc. 1st ACMSIAM Symposium on Discrete Algorithms, 1990] and imply a special tree decomposition for quadrangulations. In particular, we show how to embed quadrangulations on a 2book, so that each page contains a noncrossing alternating tree.