42 research outputs found

    Even circuits of prescribed clockwise parity

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    We show that a graph has an orientation under which every circuit of even length is clockwise odd if and only if the graph contains no subgraph which is, after the contraction of at most one circuit of odd length, an even subdivision of K_{2,3}. In fact we give a more general characterisation of graphs that have an orientation under which every even circuit has a prescribed clockwise parity. This problem was motivated by the study of Pfaffian graphs, which are the graphs that have an orientation under which every alternating circuit is clockwise odd. Their significance is that they are precisely the graphs to which Kasteleyn's powerful method for enumerating perfect matchings may be applied

    Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs

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    A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching width since its introduction. For the restricted case of bipartite graphs, we show that perfect matching width is equivalent to directed treewidth and thus the Directed Grid Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's conjecture.Comment: Manuscrip

    Pfaffian Correlation Functions of Planar Dimer Covers

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    The Pfaffian structure of the boundary monomer correlation functions in the dimer-covering planar graph models is rederived through a combinatorial / topological argument. These functions are then extended into a larger family of order-disorder correlation functions which are shown to exhibit Pfaffian structure throughout the bulk. Key tools involve combinatorial switching symmetries which are identified through the loop-gas representation of the double dimer model, and topological implications of planarity.Comment: Revised figures; corrected misprint

    Nash equilibria, gale strings, and perfect matchings

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    This thesis concerns the problem 2-NASH of ļ¬nding a Nash equilibrium of a bimatrix game, for the special class of so-called ā€œhard-to-solveā€ bimatrix games. The term ā€œhardto-solveā€ relates to the exponential running time of the famous and often used Lemkeā€“ Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings deļ¬ned by the ā€œGale evenness conditionā€ that characterise the vertices of these polytopes. We deļ¬ne the combinatorial problem ā€œAnother completely labeled Gale stringā€ whose solutions deļ¬ne the Nash equilibria of any game deļ¬ned by cyclic polytopes, including the games where the Lemkeā€“Howson algorithm takes exponential time. We show that ā€œAnother completely labeled Gale stringā€ is solvable in polynomial time by a reduction to the ā€œPerfect matchingā€ problem in Euler graphs. We adapt the Lemkeā€“Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemkeā€“Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in ļ¬nding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difļ¬culties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful
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