Noncrossing subgraphs in topological layouts
SIAM Journal on Discrete Mathematics
String graphs. I.: the number of critical nonstring graphs is infinite
Journal of Combinatorial Theory Series B
String graphs. II.: Recognizing string graphs is NP-hard
Journal of Combinatorial Theory Series B
String graphs requiring exponential representations
Journal of Combinatorial Theory Series B
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
A linear-time algorithm for drawing a planar graph on a grid
Information Processing Letters
On the rectilinear crossing number of complete graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Crossing Numbers of Graphs, Lower Bound Techniques
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
Recognizing string graphs in NP
Journal of Computer and System Sciences - STOC 2002
Journal of Computer and System Sciences - STOC 2001
Simultaneous graph embeddings with fixed edges
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
On edges crossing few other edges in simple topological complete graphs
GD'05 Proceedings of the 13th international conference on Graph Drawing
Facets in the Crossing Number Polytope
SIAM Journal on Discrete Mathematics
Complexity of some geometric and topological problems
GD'09 Proceedings of the 17th international conference on Graph Drawing
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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 R⊆(E 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.