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A linear time algorithm for embedding graphs in an arbitrary surface
 SIAM J. Discrete Math
, 1999
"... Ljubljana, February 2, 2009A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth ..."
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Cited by 56 (10 self)
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Ljubljana, February 2, 2009A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 20 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Extended Graph Rotation Systems as a Model for Cyclic Weaving on Orientable Surfaces
"... Abstract. Our interest in computer graphics motivates a revised study of graph rotation systems, which are constructs by which topological graph theorists specify the placement of graphs on surfaces, nonorientable as well as orientable. Whereas orientable imbeddings have been studied extensively, n ..."
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Cited by 4 (4 self)
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Abstract. Our interest in computer graphics motivates a revised study of graph rotation systems, which are constructs by which topological graph theorists specify the placement of graphs on surfaces, nonorientable as well as orientable. Whereas orientable imbeddings have been studied extensively, nonorientable imbeddings have been comparatively neglected. This study extends general rotation systems into a solid mathematical model for the development of an interactivegraphics cyclicweaving system. This involves a systematic exploration and characterization of dynamic surgery operations on graph rotation systems, such as edgeinsertion, edgedeletion, and edgetwisting, which are used within an interactivegraphics system for designing weavings. Some fundamental theoretical results for the model are derived. It will be seen how general rotation systems that induce nonorientable surfaces can specify weavings on orientable surfaces.
Algebraic Algorithms for Linear Matroid Parity Problems
"... We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the ma ..."
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Cited by 3 (1 self)
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We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint Spath problem, we present an O(nω)time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)time randomized algorithm where n is the number of vertices, and an O(n3)time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint Spath are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.
Topological Graph Theory  A Survey
 Cong. Num
, 1996
"... this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1 ..."
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Cited by 1 (0 self)
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this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1
Yechiam Yemini
"... this paper are a bit outofdate: not quite as good as our new results reported above!) ..."
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this paper are a bit outofdate: not quite as good as our new results reported above!)
I Artificial Intelligence
, 1996
"... ther sensing devices in a general way. A modular system that allows new sensing devices to be added incrementally would be extremely useful. 2. Multiple sensing is in many ways a problem in distributed computation. Each sensor is typically a separate computing element, with its own world model and ..."
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ther sensing devices in a general way. A modular system that allows new sensing devices to be added incrementally would be extremely useful. 2. Multiple sensing is in many ways a problem in distributed computation. Each sensor is typically a separate computing element, with its own world model and set of primitives. Each sensor has its own resolution, bandwidth and response time and a major task in robotics is getting them all to work together. An open problem is the organization of such systems; should they be hierarchical with a top level controlling "overseer" or should they be more anarchistic, with each sensor doing its own thing and a minimum level of coordination. The answer is partially task dependent, but as sensors proliferate, the solutions become more difficult. 3. Tactile sensing is becoming more important in robotics. Machine vision researchers have benefited from trying to understand biological vision systems. Robotics can also benefit from an understanding of human ha
Fundamental Cycles and Graph Embeddings 1
, 807
"... Abstract: In this paper, we investigate fundamental cycles in a graph G and their relations with graph embeddings. We show that a graph G may be embedded in an orientable surface with genus at least g if and only if for any spanning tree T, there exists a sequence of fundamental cycles C1, C2, · · ..."
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Abstract: In this paper, we investigate fundamental cycles in a graph G and their relations with graph embeddings. We show that a graph G may be embedded in an orientable surface with genus at least g if and only if for any spanning tree T, there exists a sequence of fundamental cycles C1, C2, · · ·, C2g with C2i−1 ∩ C2i ̸ = φ for 1 ≤ i ≤ g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2γM(G) cycles C1, C2, · · ·, C2γM(G) such that C2i−1 ∩ C2i ̸ = φ for 1 ≤ i ≤ γM(G), where β(G) and γM(G) are, respectively, the Betti number and the maximum genus of G. This implies that it is possible to construct an orientable embedding with large genus of a graph G from an arbitrary spanning tree T ( which may have very large number of odd components in G\E(T)). This is different from the earlier work of Xuong and Liu[9,6], where spanning trees with small odd components are needed. In fact, this makes a common generalization of Xuong[9],Liu[6] and Fu et al[2]. Further more, we show that (1).This result is useful in locating the maximum genus of a graph having a specific edgecut. Some known results for embedded graphs are also concluded;(2).The maximum genus problem may be reduced to the maximum matching problem. Based on this result and the algorithm of MicaliVazirani[8], we present a new efficient algorithm to determine the maximum genus of a graph in O((β(G)) 5 2) steps. Our method is straight and quite deferent from the algorithm of Furst,Gross and McGeoch[3] which depends on a result of Giles[4]where matroid parity method is needed.
Algebraic Algorithms in Combinatorial Optimization
"... we extend the recent algebraic approach to design fast algorithms for two problems in combinatorial optimization. First we study the linear matroid parity problem, a common generalization of graph matching and linear matroid intersection, that has applications in various areas. We show that Harvey’s ..."
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we extend the recent algebraic approach to design fast algorithms for two problems in combinatorial optimization. First we study the linear matroid parity problem, a common generalization of graph matching and linear matroid intersection, that has applications in various areas. We show that Harvey’s algorithm for linear matroid intersection can be easily generalized to linear matroid parity. This gives an algorithm that is faster and simpler than previous known algorithms. For some graph problems that can be reduced to linear matroid parity, we again show that Harvey’s algorithm for graph matching can be generalized to these problems to give faster algorithms. While linear matroid parity and some of its applications are challenging generalizations, our results show that the algebraic algorithmic framework can be adapted nicely to give faster and simpler algorithms in more general settings. Then we study the all pairs edge connectivity problem for directed graphs, where we would like to compute minimum st cut value between all pairs of vertices. Using a combinatorial approach it is not known how to solve this problem faster
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 43 (2009), Pages 79–90 The genus distributions of 4regular digraphs
"... An embedding of an Eulerian digraph in orientable surfaces was introduced by Bonnington et al. They gave some problems which need to be further studied. One of them is whether the embedding distribution of an embeddable digraph is always unimodal. In this paper, we first introduce the method of how ..."
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An embedding of an Eulerian digraph in orientable surfaces was introduced by Bonnington et al. They gave some problems which need to be further studied. One of them is whether the embedding distribution of an embeddable digraph is always unimodal. In this paper, we first introduce the method of how to determine the faces and antifaces from a given rotation scheme of a digraph. The genus distributions of two new kinds of 4regular digraphs in orientable surfaces are obtained. The genus distributions of one kind of digraph are strong unimodal, which gives a partial answer to the above problem.