On the Obfuscation Complexity of Planar Graphs

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.

💡 Research Summary

The paper introduces a novel perspective on planar graph drawings by focusing on how “obscured” a drawing can become rather than how to minimize edge crossings. For a planar graph G, the authors define the obfuscation complexity obf(G) as the maximum possible number of edge crossings over all straight‑line plane drawings of G, allowing edges to intersect arbitrarily. This contrasts with the classic crossing‑number problem, which seeks the minimum number of crossings.

The authors first present a constructive algorithm that, given any planar graph G, produces a drawing with at least one‑third of the optimal number of crossings. The method assigns an arbitrary linear order to the vertices, then draws each edge as a straight segment respecting that order, thereby forcing many edges to intersect. This yields a guaranteed lower bound of obf(G)/3 in polynomial time, showing that a reasonably good approximation of the maximum crossing number is efficiently attainable.

The main theoretical contribution is a pair of asymptotic bounds for obf(G) when the minimum vertex degree δ(G) ≥ 2. For an n‑vertex planar graph they prove

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