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A new NCalgorithm for finding a perfect matching in bipartite planar and small genus graphs (Extended Abstract)
, 2000
"... It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a Ptime algorithm for the case of graphs of small genus.) H ..."
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Cited by 8 (2 self)
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It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a Ptime algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NCalgorithm for this problem. Unlike the Miller...
Graph Minors and Graphs on Surfaces
, 2001
"... Graph minors and the theory of graphs embedded in surfaces are ..."
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Cited by 8 (3 self)
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Graph minors and the theory of graphs embedded in surfaces are
Elimination of local bridges
 Math. Slovaca
, 1997
"... Let K be a subgraph of G. It is shown that if G is 3–connected modulo K then it is possible to replace branches of K by other branches joining same pairs of main vertices of K such that G has no local bridges with respect to the new subgraph K. A linear time algorithm is presented that either perfor ..."
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Cited by 8 (8 self)
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Let K be a subgraph of G. It is shown that if G is 3–connected modulo K then it is possible to replace branches of K by other branches joining same pairs of main vertices of K such that G has no local bridges with respect to the new subgraph K. A linear time algorithm is presented that either performs such a task, or finds a Kuratowski subgraph K5 or K3,3 in a subgraph of G formed by a branch e and local bridges on e. This result is needed in linear time algorithms for embedding graphs in surfaces.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 7 (5 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
Universal obstructions for embedding extension problems
"... Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded bran ..."
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Cited by 7 (6 self)
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Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded branch size up to a bounded number of “almost disjoint ” millipedes. Moreover, B contains a nice subgraph ˜ B with the following properties. If K is 2cell embedded in some surface and F is a face of K, then ˜ B admits exactly the same types of embeddings in F as B. A linear time algorithm to construct such a universal obstruction ˜ B is presented. At the same time, for every type of embeddings of ˜ B, an embedding of B ofthesametypeis determined.
Packet Recycling: Eliminating Packet Losses due to Network Failures
"... This paper presents Packet Recycling (PR), a technique that takes advantage of cellular graph embeddings to reroute packets that would otherwise be dropped in case of link or node failures. The technique employs only one bit in the packet header to cover any single link failures, and in the order o ..."
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Cited by 6 (0 self)
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This paper presents Packet Recycling (PR), a technique that takes advantage of cellular graph embeddings to reroute packets that would otherwise be dropped in case of link or node failures. The technique employs only one bit in the packet header to cover any single link failures, and in the order of log2 (d) bits to cover all nondisconnecting failure combinations, where d is the diameter of the network. We show that our routing strategy is effective and that its path length stretch is acceptable for realistic topologies. The packet header overhead incurred by PR is very small, and the extra memory and packet processing time required to implement it at each router are insignificant. This makes PR suitable for losssensitive, missioncritical network applications. Categories and Subject Descriptors
Approximating the Crossing Number of Toroidal Graphs
"... CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time constant approximation algorithm is known for this NPcomplete problem. We prove that a natural approach to planar drawing of toroidal ..."
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CrossingNumber is one of the most challenging algorithmic problems in topological graph theory, with applications to graph drawing and VLSI layout. No polynomial time constant approximation algorithm is known for this NPcomplete problem. We prove that a natural approach to planar drawing of toroidal graphs (used already by Pach and Tóth in [20]) gives a polynomial time constant approximation algorithm for the crossing number of toroidal graphs with bounded degree. In this proof we present a new “grid” theorem on toroidal graphs.
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 5 (2 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Maximum matching in graphs with an excluded minor
 Proceedings of the Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA) 108–117
, 2007
"... Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the ..."
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Cited by 5 (5 self)
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Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the previous O(n1.5) time bound obtained by applying the O(mn1/2)time algorithm of Micali and Vazirani on thisimportant class of graphs. For graphs with bounded genus, which are special cases of Hminor free graphs, we present a randomized algorithm for finding a maximum matching in O(n!/2) < O(n1.19) time. This extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching ina planar graphs. We also present a deterministic algorithm with arunning time of O(n1+!/2) < O(n2.19) for counting thenumber of perfect matchings in graphs with bounded genus. This algorithm combines the techniques usedby the algorithms above with the counting technique of Kasteleyn. Using this algorithm we can also count,within the same running time, the number of Tjoinsin planar graphs. As special cases, we get algorithms for counting Eulerian subgraphs (T = OE) and oddsubgraphs ( T = V) of planar graphs. 1 Introduction A matching in a graph is a set of pairwise disjointedges. A perfect matching in a graph with n verticesis a matching of size n/2, and a maximum matchingis a matching of largest possible size. The problems
Isomorphism for graphs of bounded feedback vertex set number
, 2009
"... This paper presents an O(n 2) algorithm for deciding isomorphism of graphs that have bounded feedback vertex set number. This number is defined as the minimum number of vertex deletions required to obtain a forest. Our result implies that Graph Isomorphism is fixedparameter tractable with respect to ..."
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This paper presents an O(n 2) algorithm for deciding isomorphism of graphs that have bounded feedback vertex set number. This number is defined as the minimum number of vertex deletions required to obtain a forest. Our result implies that Graph Isomorphism is fixedparameter tractable with respect to the feedback vertex set number. Central to the algorithm is a new technique consisting of an application of reduction rules that produce an isomorphisminvariant outcome, interleaved with the creation of increasingly large partial isomorphisms. 1