The Complexity of Several Realizability Problems for Abstract Topological Graphs
Kyncl, Jan (2008) The Complexity of Several Realizability Problems for Abstract Topological Graphs. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 137-158 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_16).
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An $abstract topological graph$ (briefly an $AT-graph$) is a pair $A=(G,R)$ where $G=(V,E)$ is a graph and $Rsubseteq E choose 2$ is a set of pairs of its edges. An AT-graph $A$ is $simply realizable$ if $G$ can be drawn in the plane in such a way that each pair of edges from $R$ crosses exactly once and no other pair crosses. We present a polynomial algorithm which decides whether a given complete AT-graph is simply realizable. On the other hand, we show that other similar realizability problems for (complete) AT-graphs are NP-hard.
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