Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: $K_5$ and $K_{3,3}$ (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the M\"obius Ladder $V_{10}$; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of $V_{10}$; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of $V_{8}$.Comment: 176 pages, 28 figure

O natančnosti nekaterih rezultatov, ki povezujejo prereze in prekrižna števila

It is already known that for very small edge cuts in graphs, the crossing number of the graph is at least the sum of the crossing number of (slightly augmented) components resulting from the cut. Under stronger connectivity condition in each cut component that was formalized as a graph operation called zip product, a similar result was obtained for edge cuts of any size, and a natural question was asked, whether this stronger condition is necessary. In this paper, we prove that the relaxed condition is not sufficient when the size of the cut is at least four, and we prove that the gap can grow quadratically with the cut size.Znano je, da je za majhne povezavne prereze prekrižno število grafa večje ali enako vsoti prekrižnih števil nekoliko dopolnjenih komponent, ki nastanejo ob prerezu. Ob močnejših predpostavkah povezanosti vsake od komponent, ki je bilo formalizirano kot grafovska operacija \u27šiv\u27, pa lahko podoben rezultat pokažemo za povezavne prereze poljubne velikosti. Zastavi se naravno vprašanje, ali je ta pogoj potreben. V tem prispevku pokažemo, da šibkejše zahteve za povezanost komponent ne zadoščajo, če prerez vsebuje vsaj štiri povezave. Razlika med vsoto prekrižnih števil komponent in skupnega grafa lahko narašča kvadratično z velikostjo prereza