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 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.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview
of a related work is adde