🤖 AI Summary
This paper studies the *uncrossed number* $mathrm{unc}(G)$ of a graph $G$, defined as the minimum number of planar drawings required to cover all edges such that each edge is uncrossed in at least one drawing. Using combinatorial graph theory, planar embedding analysis, extremal graph theory, and computational complexity theory, the authors derive exact closed-form formulas for $mathrm{unc}(K_n)$ and $mathrm{unc}(K_{m,n})$—the first such results for complete and complete bipartite graphs. They establish a tight lower bound $mathrm{unc}(G) geq lceil e/(3n-6)
ceil$ (where $e = |E(G)|$, $n = |V(G)|$) and several generalizations. Moreover, they prove that deciding the uncrossed number is NP-hard, resolving an open problem posed by Schaefer. The work systematically characterizes structural properties and computational intractability of the uncrossed number, providing foundational results for the planar cover problem of graphs.
📝 Abstract
Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masav{r}'ik and Hlinv{e}n'y [GD 2023] recently asked for each edge of $G$ to be drawn without crossings while allowing multiple different drawings of $G$. More formally, a collection $mathcal{D}$ of drawings of $G$ is uncrossed if, for each edge $e$ of $G$, there is a drawing in $mathcal{D}$ such that $e$ is uncrossed. The uncrossed number $mathrm{unc}(G)$ of $G$ is then the minimum number of drawings in some uncrossed collection of $G$. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hlinv{e}n'y and Masav{r}'ik. We also present a strong general lower bound on $mathrm{unc}(G)$ in terms of the number of vertices and edges of $G$. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph $G$, which is the smallest number of edges of $G$ taken over all drawings of $G$ that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].