Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F22%3A43964092" target="_blank" >RIV/49777513:23520/22:43964092 - isvavai.cz</a>
Result on the web
<a href="https://www.sciencedirect.com/science/article/pii/S0095895622000077" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0095895622000077</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jctb.2022.01.007" target="_blank" >10.1016/j.jctb.2022.01.007</a>

Result language
angličtina
Original language name
Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs
Original language description
Edge-elimination is an operation of removing an edge of a cubic graph together with its endvertices and suppressing the resulting 2-valent vertices. We study the effect of this operation on the cyclic connectivity of a cubic graph. Disregarding a small number of cubic graphs with no more than six vertices, this operation cannot decrease cyclic connectivity by more than two. We show that apart from three exceptional graphs (the cube, the twisted cube, and the Petersen graph) every 2-connected cubic graph on at least eight vertices contains an edge whose elimination decreases cyclic connectivity by at most one. The proof reveals an unexpected behaviour of connectivity 6, which requires a detailed structural analysis featuring the Isaacs flower snarks and their natural generalisation, the twisted Isaacs graphs, as forced structures. A complete characterisation of this family, which includes the Heawood graph as a sporadic case, serves as the main tool for excluding the existence of exceptional graphs in connectivity 6. As an application we show that every cyclically 5-edge-connected cubic graph has a decycling set of vertices whose removal leaves a tree and the set itself has at most one edge between its vertices. This strengthens a classical result of Payan and Sakarovitch (1975) .
Czech name
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Czech description
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Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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