A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.

We examine the vertex deletion problem for weighted directed acyclic graphs (wdags). The objective is to delete the fewest number of vertices so that the resulting wdag has no path of length > d. Several simplified versions of this problem are shown to be NP-hard. However, the problem is solved in linear time when the wdag is a rooted tree and in quadratic time when the wdag is a series-parallel graph.

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We consider the family of graph problems called node-deletion problems, defined as follows: For a fixed graph property l7, what is the minimum number of nodes which must be deleted from a given graph so that the resulting subgraph satisfies l7? We show that if l7 is nontrivial and hereditary on induced subgraphs, then the node-deletion problem for n is NP-complete for both undirected and directed graphs.