Journal of Combinatorial Theory Series A
Moving vertices to make drawings plane
GD'07 Proceedings of the 15th international conference on Graph drawing
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Untangling planar graphs from a specified vertex position-Hard cases
Discrete Applied Mathematics
On collinear sets in straight-line drawings
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
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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 (@d(G)^2/24-o(1))n^2@?obf(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 @d(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 @d(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.