SIAM Journal on Computing
Improved approximations of crossings in graph drawings
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Computing crossing numbers in quadratic time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
An Algorithm for Finding Large Induced Planar Subgraphs
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
SIAM Journal on Discrete Mathematics
Crossing number is hard for cubic graphs
Journal of Combinatorial Theory Series B
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the crossing number of almost planar graphs
GD'06 Proceedings of the 14th international conference on Graph drawing
Algorithms for the hypergraph and the minor crossing number problems
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Experiments on exact crossing minimization using column generation
WEA'06 Proceedings of the 5th international conference on Experimental Algorithms
A branch-and-cut approach to the crossing number problem
Discrete Optimization
Approximating the Crossing Number of Apex Graphs
Graph Drawing
Approximating the crossing number of graphs embeddable in any orientable surface
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A tighter insertion-based approximation of the crossing number
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Vertex insertion approximates the crossing number of apex graphs
European Journal of Combinatorics
Advances in the planarization method: effective multiple edge insertions
GD'11 Proceedings of the 19th international conference on Graph Drawing
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We consider the problem of computing a crossing minimum drawing of a given planar graph G = (V, E) augmented by a star, i.e., an additional vertex v together with its incident edges Ev = {(v, u) | u ∈ V}, in which all crossings involve Ev. Alternatively, the problem can be stated as finding a planar embedding of G, in which the given star can be inserted requiring the minimum number of crossings. This is a generalization of the crossing minimum edge insertion problem [15], and can help to find improved approximations for the crossing minimization problem. Indeed, in practice, the algorithm for the crossing minimum edge insertion problem turned out to be the key for obtaining the currently strongest approximate solutions for the crossing number of general graphs. The generalization considered here can lead to even better solutions for the crossing minimization problem. Furthermore, it offers new insight into the crossing number problem for almost-planar and apex graphs. It has been an open problem whether the star insertion problem is polynomially solvable. We give an affirmative answer by describing the first efficient algorithm for this problem. This algorithm uses the SPQR-tree data structure to handle the exponential number of possible embeddings, in conjunction with dynamic programming schemes for which we introduce partitioning cost subproblems.