Curious crossing-critical edges -- variations on an example of Širáň

TL;DR

This work investigates the relationship between crossing-critical edges and Kuratowski subgraphs. By adapting S̆irán̆’s example to three graphs $G_1$, $G_2$, $G_3$ that share a common underlying simple graph $G_0$ and tuning edge multiplicities, it produces counterexamples to three natural questions about how crossings interact with Kuratowski obstructions. The authors develop a framework based on good drawings and the weighted crossing number $cr_w$, anchored by the underlying graph $G_0$ and Whitney-type arguments, to precisely determine the crossing numbers: $cr(G_1)=7$, $cr(G_2)=1$, and $cr(G_3)=21$, with corresponding properties for edges $e$ and $f$ in optimal drawings. The results show that a graph can have an edge crossed in every optimal drawing without belonging to any Kuratowski subgraph, an edge contained in all Kuratowski subgraphs but never crossed in an optimal drawing, and a crossing-critical edge that is neither in a Kuratowski subgraph nor crossed in any optimal drawing, with all phenomena realizable in simple graphs.

Abstract

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we present the following examples: a graph with an edge that is crossed in every optimal drawing of the graph, but the edge is not in any Kuratowski subgraph of the graph; a graph with an edge that is in every Kuratowski subgraph but is not crossed in any optimal drawing of the graph; and a graph with a crossing-critical edge that is not present in any Kuratowski subgraph and is not crossed in any optimal drawing of the graph.

Figures (3)

• Figure 1: S̆irán̆'s example siran: $\operatorname{cr}(G^*-e)<\operatorname{cr}(G^*)=6$, $e$ is not in any Kuratowski subgraph of $G^*$, but is crossed in the optimal drawing shown.
• Figure 2: The graphs $G_1$ and $G_2$ and a representation of $G_3$. $\ell_1=8$, $\ell_2=2$ and $\ell_3=22$.
• Figure 3: The drawing $\mathcal{D}_4$ of $G_0$.

Theorems & Definitions (9)

• Claim 1
• Claim 2
• proof
• Claim 3
• proof
• Claim 4
• proof
• Claim 5
• proof