Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms (Invited Talk)
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| Titel: | Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms (Invited Talk) |
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| Autoren: | Fomin, Fedor V., Golovach, Petr A., Sagunov, Danil, Simonov, Kirill |
| Weitere Verfasser: | Fedor V. Fomin and Petr A. Golovach and Danil Sagunov and Kirill Simonov |
| Verlagsinformationen: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022. |
| Publikationsjahr: | 2022 |
| Schlagwörter: | Erdős-Gallai theorem, dense graph, above guarantee parameterization, fixed-parameter tractability, average degree, Longest path, longest cycle, ddc:004, Dirac theorem |
| Beschreibung: | We discuss recent algorithmic extensions of two classic results of extremal combinatorics about long paths in graphs. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree d > 1, is either Hamiltonian or contains a cycle of length at least 2d. Second, the theorem of Erdős-Gallai from 1959, states that a graph G with the average vertex degree D > 1, contains a cycle of length at least D. The proofs of these theorems are constructive, they provide polynomial-time algorithms constructing cycles of lengths 2d and D. We extend these algorithmic results by showing that each of the problems, to decide whether a 2-connected graph contains a cycle of length at least 2d+k or of a cycle of length at least D+k, is fixed-parameter tractable parameterized by k. |
| Publikationsart: | Conference object Article |
| Dateibeschreibung: | application/pdf |
| Sprache: | English |
| DOI: | 10.4230/lipics.mfcs.2022.1 |
| Zugangs-URL: | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.1 |
| Rights: | CC BY |
| Dokumentencode: | edsair.dedup.wf.002..a751a47a71eddf9eccf4a18f4aeac09e |
| Datenbank: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | We discuss recent algorithmic extensions of two classic results of extremal combinatorics about long paths in graphs. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree d > 1, is either Hamiltonian or contains a cycle of length at least 2d. Second, the theorem of Erdős-Gallai from 1959, states that a graph G with the average vertex degree D > 1, contains a cycle of length at least D. The proofs of these theorems are constructive, they provide polynomial-time algorithms constructing cycles of lengths 2d and D. We extend these algorithmic results by showing that each of the problems, to decide whether a 2-connected graph contains a cycle of length at least 2d+k or of a cycle of length at least D+k, is fixed-parameter tractable parameterized by k. |
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| DOI: | 10.4230/lipics.mfcs.2022.1 |