Inserting a vertex into a planar graph


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.

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