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Efficient Algorithms for Petersen's Matching Theorem
, 1999
"... Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general gra ..."
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Cited by 24 (3 self)
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Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3regular graphs. We have developed an O(n log^4 n)time algorithm for perfect matching in a 3regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.
Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles
, 2000
"... Let G = (V, E) be a bipartite graph embedded in a plane (or nholed torus). Two subgraphs of G differ by a Ztransformation if their symmetric difference consists of the boundary edges of a single face—and if each subgraph contains an alternating set of the edges of that face. For a given φ: V ↦ → Z ..."
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Cited by 5 (1 self)
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Let G = (V, E) be a bipartite graph embedded in a plane (or nholed torus). Two subgraphs of G differ by a Ztransformation if their symmetric difference consists of the boundary edges of a single face—and if each subgraph contains an alternating set of the edges of that face. For a given φ: V ↦ → Z +, Sφ is the set of subgraphs of G in which each v ∈ V has degree φ(v). Two elements of Sφ are said to be adjacent if they differ by a Ztransformation. We determine the connected components of Sφ and assign a height function to each of its elements. If φ is identically two, and G is a grid graph, Sφ contains the partitions of the vertices of G into cycles. We prove that we can always apply a series of Ztransformations to decrease the total number of cycles provided there is enough “slack ” in the corresponding height function. This allows us to determine in polynomial time the minimal number of cycles into which G can be partitioned provided G has a limited number of nonsquare faces. In particular, we determine the Hamiltonicity of polyomino graphs in polynomial time, thereby solving a problem posed by Itai, Christos, and Szwarcfiter in 1982 [21]. The algorithm extends to nholedtorusembedded graphs that have gridlike properties. We also provide Markov chains for sampling and approximately counting the Hamiltonian cycles of G.
Computing large matchings fast
 TRANSACTIONS ON ALGORITHMS
"... In this paper we present algorithms for computing large matchings in 3regular graphs, graphs with maximum degree 3, and 3connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the bestkn ..."
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Cited by 2 (1 self)
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In this paper we present algorithms for computing large matchings in 3regular graphs, graphs with maximum degree 3, and 3connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the bestknown algorithm for computing maximum matchings in general graphs, which runs in O ( √ nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible. We also investigate graphs with block trees of bounded degree, where the dblock tree is the adjacency graph of the dconnected components of the given graph. In 3regular graphs and 3connected planar graphs with boundeddegree 2 and 4block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.
Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems
"... It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum c ..."
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Cited by 1 (0 self)
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It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number.
Computing Large Matchings in Planar Graphs with Fixed Minimum Degree
, 2009
"... In this paper we present algorithms that compute large matchings in planar graphs with fixed minimum degree. The algorithms give a guarantee on the size of the computed matching and run in linear time. Thus they are faster than the best known algorithm for computing maximum matchings in general grap ..."
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In this paper we present algorithms that compute large matchings in planar graphs with fixed minimum degree. The algorithms give a guarantee on the size of the computed matching and run in linear time. Thus they are faster than the best known algorithm for computing maximum matchings in general graphs and in planar graphs, which run in O ( √ nm) and O(n 1.188) time, respectively. For the class of planar graphs with minimum degree 3 the bounds we achieve are known to be best possible. Further, we discuss how minimum degree 5 can be used to obtain stronger bounds on the matching size.
The Number Of Matchings Of Low Order In Hexagonal Systems
"... . A simple way to calculate the number of kmatchings, k 5, in hexagonal systems is presented. Some relations between the coefficients of the characteristic polynomial of the adjacency matrix of a hexagonal system and the number of matchings are obtained. 1. Introduction A hexagonal system is a 2 ..."
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. A simple way to calculate the number of kmatchings, k 5, in hexagonal systems is presented. Some relations between the coefficients of the characteristic polynomial of the adjacency matrix of a hexagonal system and the number of matchings are obtained. 1. Introduction A hexagonal system is a 2connected plane graph G such that every interior face of G is a regular hexagon. A kmatching (or a matching of order k) of a graph G is a set of k pairwise nonadjacent edges of G. A hexagonal system has only vertices of degree 2 or 3. Note also that each hexagonal system H is a bipartite graph. It is also easy to see that H does not contain cycles of lengths 4; 8. Let G be a hexagonal system. Throughout the paper, n will denote the number of vertices whereas m will stand for the number of edges of G. By A = fa ij g n i;j=1 we will denote the adjacency matrix of G, that is a ij = ae 0; ij = 2 E (G) 1; ij 2 E (G) : Since every hexagonal system is bipartite, coefficients of the characte...