Charles Explorer logo
🇨🇿

Structure and generation of crossing-critical graphs

Publikace na Matematicko-fyzikální fakulta |
2018

Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".Abstrakt

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs.

It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs.

On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph.

This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph.