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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

The perfect matching index of a cubic graph G, denoted by π(G), is the smallest number of perfect matchings needed to cover all the edges of G; it is correctly defined for every bridgeless cubic graph. The value of π(G) is always at least 3, and if G has no 3-edge-colouring, then π(G) ≥ 4. On the other hand, a long-standing conjecture of Berge suggests that π(G) never exceeds 5. It was proved by Esperet and Mazzuoccolo [J. Graph Theory 77 (2014), 144-157] that it is NP-complete to decide for a 2-connected cubic graph whether π(G) ≤ 4. A disadvantage of the proof (noted by the authors) is that the constructed graphs have 2-cuts. We show that small cuts can be avoided and that the problem remains NP-complete even for nontrivial snarks - cyclically 4-edge-connected cubic graphs of girth at least 5 with no 3-edge-colouring. Our proof significantly differs from the one due to Esperet and Mazzuoccolo in that it combines nowhere-zero flow methods with elements of projective geometry, without referring to perfect matchings explicitly.

Martin Škoviera and Peter Varša. NP-Completeness of Perfect Matching Index of Cubic Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{skoviera_et_al:LIPIcs.STACS.2022.56, author = {\v{S}koviera, Martin and Var\v{s}a, Peter}, title = {{NP-Completeness of Perfect Matching Index of Cubic Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {56:1--56:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.56}, URN = {urn:nbn:de:0030-drops-158667}, doi = {10.4230/LIPIcs.STACS.2022.56}, annote = {Keywords: cubic graph, edge colouring, snark, perfect matching, covering, NP-completeness} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

The maximum genus gamma_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least gamma_M(G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong's combinatorial characterisation of maximum genus.

Michal Kotrbcík and Martin Skoviera. Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 14:1-14:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kotrbcik_et_al:OASIcs.SOSA.2019.14, author = {Kotrbc{\'\i}k, Michal and Skoviera, Martin}, title = {{Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {14:1--14:9}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.14}, URN = {urn:nbn:de:0030-drops-100409}, doi = {10.4230/OASIcs.SOSA.2019.14}, annote = {Keywords: maximum genus, embedding, graph, greedy algorithm} }

Document

**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

The perfect matching index of a cubic graph G, denoted by π(G), is the smallest number of perfect matchings needed to cover all the edges of G; it is correctly defined for every bridgeless cubic graph. The value of π(G) is always at least 3, and if G has no 3-edge-colouring, then π(G) ≥ 4. On the other hand, a long-standing conjecture of Berge suggests that π(G) never exceeds 5. It was proved by Esperet and Mazzuoccolo [J. Graph Theory 77 (2014), 144-157] that it is NP-complete to decide for a 2-connected cubic graph whether π(G) ≤ 4. A disadvantage of the proof (noted by the authors) is that the constructed graphs have 2-cuts. We show that small cuts can be avoided and that the problem remains NP-complete even for nontrivial snarks - cyclically 4-edge-connected cubic graphs of girth at least 5 with no 3-edge-colouring. Our proof significantly differs from the one due to Esperet and Mazzuoccolo in that it combines nowhere-zero flow methods with elements of projective geometry, without referring to perfect matchings explicitly.

Martin Škoviera and Peter Varša. NP-Completeness of Perfect Matching Index of Cubic Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{skoviera_et_al:LIPIcs.STACS.2022.56, author = {\v{S}koviera, Martin and Var\v{s}a, Peter}, title = {{NP-Completeness of Perfect Matching Index of Cubic Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {56:1--56:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.56}, URN = {urn:nbn:de:0030-drops-158667}, doi = {10.4230/LIPIcs.STACS.2022.56}, annote = {Keywords: cubic graph, edge colouring, snark, perfect matching, covering, NP-completeness} }

Document

**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

The maximum genus gamma_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least gamma_M(G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong's combinatorial characterisation of maximum genus.

Michal Kotrbcík and Martin Skoviera. Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 14:1-14:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{kotrbcik_et_al:OASIcs.SOSA.2019.14, author = {Kotrbc{\'\i}k, Michal and Skoviera, Martin}, title = {{Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {14:1--14:9}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.14}, URN = {urn:nbn:de:0030-drops-100409}, doi = {10.4230/OASIcs.SOSA.2019.14}, annote = {Keywords: maximum genus, embedding, graph, greedy algorithm} }

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