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 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 ( δ ( G ) 2 / 24 − o ( 1 ) ) n 2 ≤ obf ( G ) < 3 n 2 for an n -vertex planar graph G with minimum vertex degree δ ( G ) ≥ 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 δ ( G ) ≥ 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 δ ( G ) ≥ 2 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.