Structure and generation of crossing-critical graphs

TL;DR

This work addresses the structure and enumeration of $c$-crossing-critical graphs, the minimal graphs forcing at least $cr(G)\ge c$ edge crossings. It develops a tile-based, band/fan decomposition in plane drawings, supported by path-width and nest-depth bounds, and proves a structural theorem: large crossing-critical graphs are assembled from a finite set of bounded-size basics via expansions that replicate bounded components within long bands or fans. A key methodological blend—frames, linked/frame decompositions, Simon’s factorization forest, and semigroup composition—yields a constructive enumeration algorithm that outputs all $2$-connected $c$-crossing-critical graphs with at most $n$ vertices in time linear in the output size per graph. The results illuminate how high crossing numbers in sparse graphs arise from repeated, bounded-size building blocks and provide a foundation for algorithmic advances in crossing-number problems and related graph-minor-style analyses.

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.

Figures (5)

• Figure 1: A schematic illustration of two basic methods of constructing crossing-critical graphs. (a) The classical Möbius-twist construction by Kochol kochol1987construction; note that the ends of the plane strip are joined together in a twisted way. (b) An example construction in which the ends of a plane strip are joined together without a twist, but then a few added edges are forced to cross the strip.
• Figure 2: An illustration of Definition \ref{['def:nests']}: a $1$-nest, a $2$-nest, a proper $F$-nest, and a degenerate $F$-nest, each of depth $6$.
• Figure 3: An example of paths $P_1,\dots,P_6$ (bold lines) forming a band of length $6$ with bottom face $F_1$ and top face $F_2$, see Definition \ref{['def:band-fan']}. The five tiles of this band, as in Definition \ref{['def:tiles-support']}, are shaded in gray and the dashed arcs represent $\alpha_i$ and $\alpha_i'$ from that definition.
• Figure 4: An illustration of the double surgery from the proof of Lemma \ref{['lemma-redurem']}.
• Figure 5: An illustration of the surgeries from the proof of Lemma \ref{['lemma-exprem']}.

Theorems & Definitions (84)

• Definition 1.1: crossing-critical
• Definition 2.1: crossing number
• Theorem 2.2: Richter and Thomassen richter1993minimal
• Proposition 2.3: folklore
• proof
• proof
• Definition 2.5: path decomposition and path-width
• Theorem 2.6: Hliněný Hl
• Definition 2.7: nests
• Theorem 2.8: Hernandez-Velez et al. crnest