On the obfuscation complexity of planar graphs

Abstract

Being motivated by John Tantalo’s Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $\mathrm{obf}\left(G\right)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $\mathrm{obf}\left(G\right)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $\mathrm{obf}\left(G\right)/3$ edge crossings. We prove bounds $(\delta {\left(G\right)}^2/24-o\left(1\right))n^2\le \mathrm{obf}\left(G\right)<3n^2$ for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta \left(G\right)\ge 2$.

The shift complexity of $G$, denoted by $\mathrm{shift}\left(G\right)$, is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of $G$ (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If $\delta \left(G\right)\ge 3$, then $\mathrm{shift}\left(G\right)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta \left(G\right)\ge 2$ we notice that $\mathrm{shift}\left(G\right)$ can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing $D$ of a planar graph, it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.