🤖 AI Summary
This paper investigates the structural characterization and efficient generation of *c*-crossing-critical graphs—minimal graphs requiring exactly *c* edge crossings in any planar embedding. For *c* > 1, we establish a breakthrough structural result: all *c*-crossing-critical graphs have bounded pathwidth. We introduce the “narrow-band/fan replication” model—a unified framework capturing the local repetitive structure underlying infinitely many families of such graphs. We prove that every *c*-crossing-critical graph can be generated from a seed graph of bounded size via localized replications. Leveraging this characterization, we design the first polynomial-delay enumeration algorithm that outputs all *n*-vertex *c*-crossing-critical graphs in polynomial time per graph. Our work resolves two longstanding bottlenecks in crossing number theory: the lack of precise structural understanding and the absence of controlled constructive methods. The results provide a new paradigm for both structural graph theory and algorithmic graph drawing.
📝 Abstract
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.
