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Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
The directed steiner network problem is tractable for a constant number of terminals
 In Proceedings FOCS
, 1999
"... We consider the DIRECTED STEINER NETWORK problem, also called the POINTTOPOINT CONNECTION problem, where given a directed graph G and p pairs s1 t1 sp tp of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from si to ti for all i. The problem is NPhard for gene ..."
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We consider the DIRECTED STEINER NETWORK problem, also called the POINTTOPOINT CONNECTION problem, where given a directed graph G and p pairs s1 t1 sp tp of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from si to ti for all i. The problem is NPhard for general p, since the DIRECTED STEINER TREE problem is a special case. Until now, the complexity was unknown for constant p 3. We prove that the problem is polynomially solvable if p is any constant number, even if nodes and edges in G are weighted and the goal is to minimize the total weight of the subgraph H. In addition, we give an efficient algorithm for the STRONGLY CONNECTED STEINER SUBGRAPH problem for any constant p, where given a directed graph and p nodes in the graph, one has to compute the smallest strongly connected subgraph containing the p nodes.
Upward Embeddings and Orientations of Undirected Planar Graphs
 Journal of Graph Algorithms and Applications
, 2001
"... An upward embedding of an embedded planar graph states, for each vertex v, which edges are incident to v \above" or \below" and, in turn, induces an upward orientation of the edges. In this paper we characterize the set of all upward embeddings and orientations of a plane graph by using a ..."
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An upward embedding of an embedded planar graph states, for each vertex v, which edges are incident to v \above" or \below" and, in turn, induces an upward orientation of the edges. In this paper we characterize the set of all upward embeddings and orientations of a plane graph by using a simple ow model. We take advantage of such a ow model to compute upward orientations with the minimum number of sources and sinks of 1connected graphs. Our theoretical results allow us to easily compute visibility representations of 1connected graphs while having a certain control over the width and the height of the computed drawings, and to deal with partial assignments of the upward embeddings \underlying" the visibility representations. 2 1
Finding Minimum Congestion Spanning Trees
"... Abstract. Given a graph G and a positive integer k, we want to nd k spanning trees on G, not necessarily disjoint, of minimum total weight, such that the weight of each edge is subject to a penalty function if it belongs to more than one tree. We present a polynomial time algorithm for this problem; ..."
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Abstract. Given a graph G and a positive integer k, we want to nd k spanning trees on G, not necessarily disjoint, of minimum total weight, such that the weight of each edge is subject to a penalty function if it belongs to more than one tree. We present a polynomial time algorithm for this problem; the algorithm's complexity is quadratic in k. We also present two heuristics with complexity linear in k. In an experimental study we show that these heuristics are much faster than the exact algorithm also in practice, and that their solutions are around 1 % of optimal for small values of k and much better for large k. 1