30 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

    Charakterisierung Pfaff’scher Graphen mittels verbotener Teilgraphen

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    Pfaff’sche Graphen sind genau jene, auf die man Kasteleyns Methode zum Abzählen perfekter Matchings anwenden kann, womit dieses Problem in polynomieller Zeit lösbar ist. Diese Arbeit soll darlegen, für welche Klassen von Graphen eine einfache und schöne Charakterisierung Pfaff’scher Graphen existiert. Das Problem ergibt sich aus der unhandlichen und nur umständlich zu überprüfenden Definition. Dabei wird insbesondere die Charakterisierung mittels verbotener Teilgraphen im Mittelpunkt stehen. Die Idee ist, eine Liste von (möglichst wenigen) Graphen anzugeben, deren “nicht-enthalten-Sein” als sogenannter „Matching Minor” eine notwendige und hinreichende Bedingung dafür darstellt, dass es sich um einen Pfaff’schen Graphen handelt.Pfaffian graphs are exactly those, on which one can apply Kasteleyns method of counting perfect matchings, which implies the polynomial-time-solvability of our problem. This paper shall demonstrate for which classes of graphs there exists a facile characterisation of Pfaffian graphs. The difficulties here originate in the unhandy definition, whose requirements are hard to check. Thereby, the characterisation in terms of forbidden subgraphs will be the central issue of our studies. The idea is to state a list of (preferably few) graphs, so that for a given graph, containing one of the graphs in the list as a so-called matching minor is equivalent with beeing non-Pfaffian

    Brick Generation and Conformal Subgraphs

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    A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, called bricks (which are nonbipartite and 3-connected) and braces (which are bipartite). A fundamental theorem of LovU+00E1sz (1983) states that every nonbipartite matching covered graph admits an ear decomposition starting with a bi-subdivision of K4K_4 or of the triangular prism C6‾\overline{C_6}. This led Carvalho, Lucchesi and Murty (2003) to pose two problems: (i) characterize those nonbipartite matching covered graphs which admit an ear decomposition starting with a bi-subdivision of K4K_4, and likewise, (ii) characterize those which admit an ear decomposition starting with a bi-subdivision of C6‾\overline{C_6}. In the first part of this thesis, we solve these problems for the special case of planar graphs. In Chapter 2, we reduce these problems to the case of bricks, and in Chapter 3, we solve both problems when the graph under consideration is a planar brick. A nonbipartite matching covered graph G is near-bipartite if it has a pair of edges U+03B1 and U+03B2 such that G-{U+03B1,U+03B2} is bipartite and matching covered; examples are K4K_4 and C6‾\overline{C_6}. The first nonbipartite graph in any ear decomposition of a nonbipartite graph is a bi-subdivision of a near-bipartite graph. For this reason, near-bipartite graphs play a central role in the theory of matching covered graphs. In the second part of this thesis, we establish generation theorems which are specific to near-bipartite bricks. Deleting an edge e from a brick G results in a graph with zero, one or two vertices of degree two, as G is 3-connected. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of G-e is the graph J obtained from it by bicontracting all its vertices of degree two. The edge e is thin if J is also a brick. Carvalho, Lucchesi and Murty (2006) showed that every brick, distinct from K4K_4, C6‾\overline{C_6} and the Petersen graph, has a thin edge. In general, given a near-bipartite brick G and a thin edge e, the retract J of G-e need not be near-bipartite. In Chapter 5, we show that every near-bipartite brick G, distinct from K4K_4 and C6‾\overline{C_6}, has a thin edge e such that the retract J of G-e is also near-bipartite. Our theorem is a refinement of the result of Carvalho, Lucchesi and Murty which is appropriate for the restricted class of near-bipartite bricks. For a simple brick G and a thin edge e, the retract of G-e may not be simple. It was established by Norine and Thomas (2007) that each simple brick, which is not in any of five well-defined infinite families of graphs, and is not isomorphic to the Petersen graph, has a thin edge such that the retract J of G-e is also simple. In Chapter 6, using our result from Chapter 5, we show that every simple near-bipartite brick G has a thin edge e such that the retract J of G-e is also simple and near-bipartite, unless G belongs to any of eight infinite families of graphs. This is a refinement of the theorem of Norine and Thomas which is appropriate for the restricted class of near-bipartite bricks

    EUROCOMB 21 Book of extended abstracts

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