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21
A Separator Theorem for Planar Graphs f
, 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 finds su ..."
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Cited by 397 (1 self)
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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 finds such a partition A, B, C in O(n) time.
On Infinite Cycles I
"... We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, a ..."
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Cited by 29 (11 self)
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We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the endfaithful spanning trees. We also generalize Euler's theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.
Topological Paths, Cycles and Spanning Trees in Infinite Graphs
"... We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph a ..."
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Cited by 26 (13 self)
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We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only its topological ends are added as new points, while rays from other ends are made to converge to certain vertices.
An O(m log n)Time Algorithm for the Maximal Planar Subgraph Problem
, 1993
"... Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C1 ..."
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Cited by 17 (0 self)
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Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [15], Wu defined the problem of planar graphs in terms of the following four subproblems: ################## 1 This work was partly supported by ThomsonCSF/DSE and by the National Science Foundation under grant CCR9002428. 2. Research at Princeton University partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, grant NSFSTC8809648, and the Office of Naval Research, contract N0001487K0467.    2  P1. Decide whether a connected graph G is planar. P2. Find a minimal set of edges the removal of which will render the remaining part of G planar. P3. Gi...
On Infinite Cycles II
"... We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove ..."
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Cited by 14 (0 self)
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We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edgedecomposition into single cycles and orthogonality to cuts.
Counting Embeddings of Planar Graphs Using DFS Trees
 SIAM Journal on Discrete Mathematics
, 1993
"... Previously counting embeddings of planar graphs [5] used PQ trees and was restricted to biconnected graphs. Although the PQ tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give for ..."
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Cited by 4 (0 self)
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Previously counting embeddings of planar graphs [5] used PQ trees and was restricted to biconnected graphs. Although the PQ tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give formulas that count the number of embeddings of general planar graphs (not necessarily connected or biconnected) in O (n) arithmetic steps, where n is the number of vertices of the input graph. Finally, our algorithm can be extended to generate all embeddings of a planar graph in linear time with respect to the output. Key words. graph, depth first search, embedding, planar graph, articulation point, connected component AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [14], Wu stated four basic planar graph problems: 1. Decide whether a connected graph G is planar. 2. Find a minimal set of edges the removal of which will render the remaining part of G planar. ...
The Fundamental Group of Locally Finite Graphs with Ends
, 2008
"... We characterize the fundamental group of a locally finite graph G with ends, by embedding it canonically as a subgroup in the inverse limit of the free groups ss1(G0) with G0 ` G finite. As an intermediate step, we characterize ss1(G) combinatorially as a group of infinite words. ..."
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Cited by 4 (3 self)
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We characterize the fundamental group of a locally finite graph G with ends, by embedding it canonically as a subgroup in the inverse limit of the free groups ss1(G0) with G0 ` G finite. As an intermediate step, we characterize ss1(G) combinatorially as a group of infinite words.
Topological circles and Euler tours in locally finite graphs
 ELECTRONIC J. COMB
, 2009
"... We obtain three results concerning topological paths ands circles in the end compactification G of a locally finite connected graph G. Confirming a conjecture of Diestel we show that through every edge set E ∈ C there is a topological Euler tour, a continuous map from the circle S 1 to the end com ..."
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Cited by 4 (2 self)
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We obtain three results concerning topological paths ands circles in the end compactification G of a locally finite connected graph G. Confirming a conjecture of Diestel we show that through every edge set E ∈ C there is a topological Euler tour, a continuous map from the circle S 1 to the end compactification G  of G that traverses every edge in E exactly once and traverses no other edge. Second, we show that for every sequence (τi)i∈N of topological x–y paths in G there is a topological x–y path in G  all of whose edges lie eventually in every member of some fixed subsequence of (τi). It is pointed out that this simple fact has several applications some of which reach out of the realm of G. Third, we show that every set of edges not containing a finite odd cut of G extends to an element of C.
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.