A network flow approach to a common generalization of Clar and Fries numbers
Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of these two concepts for bipartite plane graphs, and then we extend it further to the n...
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| Veröffentlicht in: | Discrete mathematics Jg. 347; H. 11; S. 114145 |
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01.11.2024
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| Abstract | Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of these two concepts for bipartite plane graphs, and then we extend it further to the notion of source-sink pairs of subsets of nodes in a general (not necessarily planar) directed graph. The main result is a min-max formula for the maximum weight of a source-sink pair. The proof is based on the recognition that the convex hull of source-sink pairs can be obtained as the projection of a network tension polyhedron. The construction makes it possible to apply any standard cheapest network flow algorithm to compute both a maximum weight source-sink pair and a minimizer of the dual optimization problem formulated in the min-max theorem. As a consequence, our approach gives rise to the first purely combinatorial, strongly polynomial algorithm to compute a largest (or even a maximum weight) Fries-set of a perfectly matchable plane bipartite graph and an optimal solution to the dual minimization problem. |
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| AbstractList | Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of these two concepts for bipartite plane graphs, and then we extend it further to the notion of source-sink pairs of subsets of nodes in a general (not necessarily planar) directed graph. The main result is a min-max formula for the maximum weight of a source-sink pair. The proof is based on the recognition that the convex hull of source-sink pairs can be obtained as the projection of a network tension polyhedron. The construction makes it possible to apply any standard cheapest network flow algorithm to compute both a maximum weight source-sink pair and a minimizer of the dual optimization problem formulated in the min-max theorem. As a consequence, our approach gives rise to the first purely combinatorial, strongly polynomial algorithm to compute a largest (or even a maximum weight) Fries-set of a perfectly matchable plane bipartite graph and an optimal solution to the dual minimization problem. |
| ArticleNumber | 114145 |
| Author | Frank, András Bérczi-Kovács, Erika |
| Author_xml | – sequence: 1 givenname: Erika orcidid: 0000-0003-2259-0868 surname: Bérczi-Kovács fullname: Bérczi-Kovács, Erika email: erika.berczi-kovacs@ttk.elte.hu organization: HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary – sequence: 2 givenname: András surname: Frank fullname: Frank, András email: andras.frank@ttk.elte.hu organization: HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization, Hungary |
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| Cites_doi | 10.1016/j.apm.2015.01.005 10.1287/opre.34.2.250 10.1021/acs.jcim.1c00735 10.1112/plms/s3-13.1.743 10.1137/110849390 10.1007/s10910-017-0799-8 10.1007/BF02020271 10.1007/s00493-007-2073-3 10.1016/j.laa.2006.07.026 10.1021/ci00018a012 10.1016/j.jctb.2006.09.005 10.1016/j.dam.2009.11.004 10.1007/BF01277551 10.1007/s10910-013-0193-0 10.1039/ft9928801621 10.1002/jlac.19274540108 10.1007/s10910-014-0305-5 |
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| Keywords | Clar number Polynomial-time algorithm Fries number |
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| Title | A network flow approach to a common generalization of Clar and Fries numbers |
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