| Title: | Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs |
| Authors: | Nedela, Roman Škoviera, Martin |
| Citation: | NEDELA, R. ŠKOVIERA, M. Cyclic connectivity, edge-elimination, and the twisted Isaacs graphs. Journal of Combinatorial Theory, Series B, 2022, roč. 155, č. Leden, s. 17-44. ISSN: 0095-8956 |
| Issue Date: | 2022 |
| Publisher: | Academic Press Inc. |
| Document type: | článek article |
| URI: | 2-s2.0-85123885552 http://hdl.handle.net/11025/47057 |
| ISSN: | 0095-8956 |
| Keywords in different language: | graph;connectivity |
| Abstract: | V práci studujeme efekt hranové eliminace na cyklickou souvislost kubického grafu. Dokážeme, že kromě tří výjimečných grafů, v kubickom grafe existuje hrana, jejíž eliminace zmenší cyklickou souvislost nejvýše o jednotku. Zvláštní chování kubických grafů souvislosti 6 vzhledem k eliminaci hran nás přivedlo k charakterizaci Issacsových grafů, které hrají důležitou roli při studiu kubických grafů. Hlavní výsledek je použitý na důkaz tvrzení, že pro každý 5-souvislý kubický graf existuje rozklad vrcholové množiny na indukovaný strom a indukovaný podgraf s nejvýše jednou hranou. Tento výsledek zobecňuje větu Payana a Sakharoviche z roku 1975. |
| Abstract in different language: | 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) . |
| Rights: | Plný text je přístupný v rámci univerzity přihlášeným uživatelům. © Elsevier |
| Appears in Collections: | Články / Articles (KMA) OBD |
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