1,337 research outputs found

    Bases for the Matching Lattice of Matching Covered Graphs

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    This article contains a brief introduction to the theory of matching covered graphs: ear-decompositions, the matching lattice and the most important results of the theory. In its last section we prove that the matching lattice of any matching covered graph has a basis consisting solely of perfect matchings and we present a conjecture relating the minimum number of double ears of any eardecomposition of a matching covered graph and the number of bricks and bricks isomorphic to the Petersen graph in any brick decomposition of the same graph

    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

    Robust Assignments via Ear Decompositions and Randomized Rounding

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    Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape
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