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A Note on Contraction Degeneracy
, 2004
"... The parameter contraction degeneracy — the maximum minimum degree over all minors of a graph — is a treewidth lower bound and was first defined in [3]. In experiments it was shown that this lower bound improves upon other treewidth lower bounds [3]. In this note, we examine some relationships betwe ..."
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The parameter contraction degeneracy — the maximum minimum degree over all minors of a graph — is a treewidth lower bound and was first defined in [3]. In experiments it was shown that this lower bound improves upon other treewidth lower bounds [3]. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, blocks of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.
Planar Graphs with Topological Constraints
 Journal of Graph Algorithms and Applications
, 2002
"... We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle... ..."
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We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, userdefined topological constraints. The constraints consist each of a cycle...
A Simple Linear Time Algorithm for Embedding Maximal Planar Graphs
, 1993
"... All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [ ..."
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All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex datastructures, or they are easy to understand and implement, but require more than linear time [2]. In this paper, a new lineartime algorithm for embedding maximal planar graphs is proposed. This algorithm is both easy to understand and easy to implement. The algorithm consists of three phases which use only simple, local graphmodifications. In addition to planar embedding, the new algorithm allows to test graphs for maximal planarity. The generation of Straight Line Drawings by a technique of Read [12] can be naturally incorporated into the algorithm. We also demonstrate how to generate random (maximal) planar graphs. The algorithm presented constitutes a first step towards a simple, lineartime solution for embedding general planar graphs. 2 3 1 Introduction One of the main top...
Graph Embedding with Topological CycleConstraints
 Graph Drawing, Proc. 7th Int. Symp. GD'99, LNCS 1731, 155164
, 1999
"... . This paper concerns graph embedding under topological constraints. We address the problem of finding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle. This problem turns out to be NPc ..."
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. This paper concerns graph embedding under topological constraints. We address the problem of finding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle. This problem turns out to be NPcomplete. However, towards an analysis of its tractable subproblems, we develop an efficient algorithm for the special case where graphs are 2connected and any two distinct cycles in the constraints have at most one vertex in common. 1 Introduction In many graph drawing applications it is important to find drawings with certain geometrical or topological properties. For instance, to support the semantics of visual languages, given subgraphs should be drawn with a predefined shape. Such userdefined requirements on the graph layout are called graph drawing constraints. Due to their wide applicability, graph drawing approaches supporting them are of recent research interest [7]. So far most o...
The LeftRight Planarity Test
, 2009
"... A graph is planar if and only if it can be embedded in the plane without crossings. I give a detailed exposition of simple and efficient, yet poorly known algorithms for planarity testing, embedding, and Kuratowski subgraph extraction based on the leftright characterization of planarity. ..."
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A graph is planar if and only if it can be embedded in the plane without crossings. I give a detailed exposition of simple and efficient, yet poorly known algorithms for planarity testing, embedding, and Kuratowski subgraph extraction based on the leftright characterization of planarity.
Contractions, Removals and How to Certify 3Connectivity in Linear Time
"... One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the exis ..."
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One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K4, such that every intermediate graph is 3vertexconnected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(V  2) to O(E). This result has a number of consequences; an important one is a new lineartime test of 3connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in the last years. The test is conceptually different from wellknown lineartime 3connectivity tests and uses a certificate that is easy to verify in time O(E). We show how to extend the results to an optimal certifying test of 3edgeconnectivity. 1
Chapter 6 Planarity Section 6.1 Euler’s Formula
"... In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities with each of the three houses so that none of the utility lines crossed. We attempted a drawing of the graph model K 3, 3 in Figure 1.1.2. I ..."
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In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities with each of the three houses so that none of the utility lines crossed. We attempted a drawing of the graph model K 3, 3 in Figure 1.1.2. In this chapter we will show that no such drawing is possible in the plane. A (p, q) graph G is said to be embeddable in the plane or planar if it is possible to draw G in the plane so that the edges of G intersect only at end vertices. If such a drawing has been done, we say that a plane embedding of the graph has been found. v 1 v 2 r 1 r 2 r 3 v 3 v 4 r 4 Figure 6.1.1. A plane embedding of K 4. Given a plane graph G, a region of G is a maximal section of the plane for which any two points may be joined by a curve. Intuitively, a region is the connected section of the plane bounded (often enclosed) by some set of edges of G. In Figure 6.1.1, the regions of G are labeled r 1, r 2, r 3 and r 4. The region r 1 is bounded by the edges v 1 v 2, v 2 v 3 and v 3 v 1, while r 2 is bounded by v 1 v 2, v 2 v 4 and v 4 v 1. Further, note that every plane graph has a region similar to the region r 1. This region, which is actually not enclosed, is called the exterior region. It is also the case that given any planar graph G, there is an embedding of G with any region as the exterior region. One of the most useful results for dealing with planar graphs relates the order, size and number of regions of the graph. This result is originally from Euler [1].