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12
The Cycle Space of an Infinite Graph
 COMB., PROBAB. COMPUT
, 2004
"... Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph togethe ..."
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Cited by 26 (9 self)
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Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach
MacLane's Planarity Criterion for Locally Finite Graphs
, 2003
"... MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner (1970), we show that the cycle space introduced recently by Diestel and Kühn allows a verbatim ge ..."
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Cited by 16 (2 self)
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MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner (1970), we show that the cycle space introduced recently by Diestel and Kühn allows a verbatim generalization of MacLane's criterion to locally finite graphs.
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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Cited by 5 (0 self)
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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...
Optimizing Over All Combinatorial Embeddings Of A Planar Graph
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, PROC. 7TH INT. IPCO CONF., LNCS 1610, 361376
, 1999
"... We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an importan ..."
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Cited by 4 (2 self)
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We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an important influence on the aesthetics of the drawing. We characterize
Graph topologies induced by edge lengths
, 2009
"... Let G be a graph each edge e of which is given a length ℓ(e). This naturally induces a distance dℓ(x, y) between any two vertices x, y, and we let Gℓ denote the completion of the corresponding metric space. It turns out that several well studied topologies on infinite graphs are special cases of  ..."
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Cited by 4 (3 self)
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Let G be a graph each edge e of which is given a length ℓ(e). This naturally induces a distance dℓ(x, y) between any two vertices x, y, and we let Gℓ denote the completion of the corresponding metric space. It turns out that several well studied topologies on infinite graphs are special cases of Gℓ. Moreover, it seems that Gℓ is the right setting for studying various problems. The aim of this paper is to introduce Gℓ, providing basic facts, motivating examples and open problems, and indicate possible applications. Parts of this work suggest interactions between graph theory and other fields, including algebraic topology and geometric group theory.
The basis number of the powers of the complete graph
 Discrete Math
, 1998
"... A basis of the cycle space C(G) of a graph G is hfold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that C(G) has an hfold basis. MacLane [3] showed that a graph G is planar if and only if b(G) ≤ 2. Schmeichel [4] proved that b(K ..."
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Cited by 3 (0 self)
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A basis of the cycle space C(G) of a graph G is hfold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that C(G) has an hfold basis. MacLane [3] showed that a graph G is planar if and only if b(G) ≤ 2. Schmeichel [4] proved that b(Kn) ≤ 3, and Banks and Schmeichel [2] proved that b(K d 2) ≤ 4 where Kd 2 is the ddimesional hypercube. We show that b(K d n) ≤ 9 for any n and d, where K d n is the cartesian dth power of the complete graph Kn. 0 Keywords: cycle space of a graph, basis number, powers of complete graphs 1
Towards a Model Theory of Figure Ground Locations
 Proc. 6th Symp. on Mathematics and AI, Fort
"... In this paper a model theory for figure ground locations is proposed. Figure ground locations are ntuples of predicates about relations between regional individuals and a set of regional individuals forming a regional partition of the plane. This language is based on the RCCtheory which is extende ..."
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Cited by 2 (0 self)
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In this paper a model theory for figure ground locations is proposed. Figure ground locations are ntuples of predicates about relations between regional individuals and a set of regional individuals forming a regional partition of the plane. This language is based on the RCCtheory which is extended by a number of definitions and axioms constraining the models to the set of regular closed sets in a 2dimensional T4 topological space. In this paper we give two models of figure ground locations: One model in point set topology referring to equivalence classes of regular closed sets. One graph theoretical model which is a set of directed and edgevalued versions of the nondirected nonlabeled dual graph of the planar graph representing the regional partition. Both models are shown to be isormorphic with respect to a set of union and intersection operations. 1. INTRODUCTION Much effort has been expended on the problem of constructing formal theories for qualitative spatial reasoning (QS...
A New Parallel Algorithm for Planarity Testing
, 2003
"... This paper presents a new parallel algorithm for planarity testing based upon the work of Klein and Reif [14]. Our new approach gives correct answers on instances that provoke false positive and false negative results using Klein and Reif's algorithm. The new algorithm has the same complexity bounds ..."
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Cited by 1 (1 self)
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This paper presents a new parallel algorithm for planarity testing based upon the work of Klein and Reif [14]. Our new approach gives correct answers on instances that provoke false positive and false negative results using Klein and Reif's algorithm. The new algorithm has the same complexity bounds as Klein and Reif's algorithm and runs in O n processors of a Concurrent Read Exclusive Write (CREW) Parallel RAM (PRAM). Implementations of the major steps of this parallel algorithm exist for symmetric multiprocessors and exhibit speedup when compared to the best sequential approach. Thus, this new parallel algorithm for planarity testing lends itself to a highperformance sharedmemory implementation.
Embedding graphs in surfaces: MacLane’s theorem for higher genus
"... Given a closed surface S, we characterise the graphs embeddable in S by an algebraic condition asserting the existence of a sparse generating set for their cycle space. When S is the sphere, the condition defaults to MacLane’s planarity criterion. ..."
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Cited by 1 (1 self)
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Given a closed surface S, we characterise the graphs embeddable in S by an algebraic condition asserting the existence of a sparse generating set for their cycle space. When S is the sphere, the condition defaults to MacLane’s planarity criterion.