Distinguishing graphs via cycles
In this paper, we employ the cycle regularity parameter to devise efficient recognition algorithms for three highly symmetric graph families: folded cubes, I-graphs, and double generalized Petersen graphs. For integers ℓ,λ,m a simple graph is [ℓ,λ,m]-cycle regular if every path of length ℓ belongs t...
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| Vydané v: | Discrete Applied Mathematics Ročník 364; s. 74 - 98 |
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Elsevier B.V
31.03.2025
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| ISSN: | 0166-218X |
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| Abstract | In this paper, we employ the cycle regularity parameter to devise efficient recognition algorithms for three highly symmetric graph families: folded cubes, I-graphs, and double generalized Petersen graphs.
For integers ℓ,λ,m a simple graph is [ℓ,λ,m]-cycle regular if every path of length ℓ belongs to exactly λ different cycles of length m. We identify all [1,λ,8]-cycle regular I-graphs and all [1,λ,8]-cycle regular double generalized Petersen graphs. For n≥7 we show that a folded cube FQn is [1,n−1,4], [1,4n2−12n+8,6] and [2,4n−8,6]-cycle regular, and identify the corresponding exceptional values of cycle regularity for n<7. As a consequence, we describe a linear recognition algorithm for double generalized Petersen graphs, an O(|E|log|V|) recognition algorithm for the family of folded cubes, and an O(|V|2) recognition algorithm for I-graphs.
We believe the structural observations and methods used in the paper are of independent interest and could be used to solve other algorithmic problems. The results of this paper have been presented at COCOON 2021. |
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| AbstractList | In this paper, we employ the cycle regularity parameter to devise efficient recognition algorithms for three highly symmetric graph families: folded cubes, I-graphs, and double generalized Petersen graphs.
For integers ℓ,λ,m a simple graph is [ℓ,λ,m]-cycle regular if every path of length ℓ belongs to exactly λ different cycles of length m. We identify all [1,λ,8]-cycle regular I-graphs and all [1,λ,8]-cycle regular double generalized Petersen graphs. For n≥7 we show that a folded cube FQn is [1,n−1,4], [1,4n2−12n+8,6] and [2,4n−8,6]-cycle regular, and identify the corresponding exceptional values of cycle regularity for n<7. As a consequence, we describe a linear recognition algorithm for double generalized Petersen graphs, an O(|E|log|V|) recognition algorithm for the family of folded cubes, and an O(|V|2) recognition algorithm for I-graphs.
We believe the structural observations and methods used in the paper are of independent interest and could be used to solve other algorithmic problems. The results of this paper have been presented at COCOON 2021. |
| Author | Krnc, Matjaž Klobas, Nina |
| Author_xml | – sequence: 1 givenname: Nina orcidid: 0000-0002-8024-5782 surname: Klobas fullname: Klobas, Nina email: nina.klobas@durham.ac.uk organization: Department of Computer Science, Durham University, United Kingdom – sequence: 2 givenname: Matjaž orcidid: 0000-0002-4960-8901 surname: Krnc fullname: Krnc, Matjaž email: matjaz.krnc@upr.si organization: FAMNIT and IAM, University of Primorska, Slovenia |
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| Cites_doi | 10.26493/1855-3974.113.3dc 10.1016/S0095-8956(81)80026-3 10.4153/CMB-1972-012-3 10.1016/S0021-9800(69)80116-X 10.1007/BF02122560 10.1016/0095-8956(91)90091-W 10.1016/S0166-218X(00)00292-4 10.1109/71.80187 10.1002/jgt.3190190102 10.1016/j.disc.2016.05.032 10.1006/jctb.1997.1729 10.1016/S0012-365X(02)00575-7 10.1007/s10479-008-0392-4 10.1007/s00373-011-1086-2 10.1016/0012-365X(79)90095-5 10.1016/S0196-6774(03)00048-8 10.1109/TIT.1983.1056664 10.37236/2087 10.1080/16073606.1993.9631741 10.1016/j.dam.2020.03.007 10.2140/pjm.1972.40.53 10.1017/S0305004100049811 10.1002/jgt.22642 10.1016/j.aml.2005.04.002 10.1002/jcd.20054 10.1090/S0002-9904-1950-09407-5 10.1016/S0925-7721(97)00014-X 10.1016/0012-365X(92)90649-Z 10.1016/j.ejc.2005.04.014 10.1016/0012-365X(91)90397-K 10.1016/0095-8956(83)90042-4 10.1016/j.camwa.2006.10.033 |
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| Keywords | Folded cubes Generalized Petersen graphs Recognition algorithm Double generalized Petersen graph |
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| Title | Distinguishing graphs via cycles |
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