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  • 1
    Electronic Resource
    Electronic Resource
    A note on packing paths in planar graphs (1995)
    Springer
    Mathematical programming 70 (1995), S. 201-209 
    Details
    ISSN: 1436-4646
    Keywords: Disjoint paths ; Joins ; Packing cuts
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Seymour (1981) proved that the cut criterion is necessary and sufficient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, Middendorf and Pfeiffer (1993) proved the problem to be NP-complete. For this case, Korach and Penn (1992) proved that the cut criterion is sufficient for the existence of a near-complete packing of paths. Here we generalize this result by showing how a natural strengthening of the cut criterion yields better packings of paths. Analogously to Seymour's approach, we actually prove a theorem on packing cuts in an arbitrary graph and then the planar edge-disjoint paths case is obtained by planar dualization. The main result is derived from a theorem of Sebő (1990) on the structure of ±1 weightings of a bipartite graph with no negative circuits.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585937
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Conservative weightings and ear-decompositions of graphs (1993)
    Springer
    Combinatorica 13 (1993), S. 65-81 
    Details
    ISSN: 1439-6912
    Keywords: 05 C 70 ; 05 C 75 ; 94 B 60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A subsetJ of edges of a connected undirected graphG=(V, E) is called ajoin if |C∩J|≤|C|/2 for every circuitC ofG. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality μ of a joint ofG. Namely, μ=(φ+|V|−1)/2 where φ denotes the minimum number of edges whose contraction leaves a factor-critical graph. To study these parameters we introduce a new decomposition ofG, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always φ and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202790
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    Paper (German National Licenses)
    Fulltext
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