On the Obfuscation Complexity of Planar Graphs

TL;DR

This paper investigates the maximum obfuscation of planar graph drawings by analyzing edge crossings and vertex shifts, providing bounds and complexity results related to graph obfuscation and crossing removal.

Contribution

It introduces the concept of obfuscation complexity for planar graphs, establishes bounds on this measure, and proves the NP-hardness of crossing removal via vertex shifts.

Findings

Bounds on obfuscation complexity for planar graphs.

Linear shift complexity for graphs with minimum degree 3.

NP-hardness of crossing elimination through vertex shifts.

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 $obf(G)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge crossings. We prove bounds $(\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2$ for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$. The shift complexity of $G$, denoted by $shift(G)$, 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 $\delta(G)\ge 3$, then $shift(G)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that $shift(G)$ 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.