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An O(m log n)Time Algorithm for the Maximal Planar Subgraph Problem
, 1993
"... Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C1 ..."
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Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depthfirstsearch, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [15], Wu defined the problem of planar graphs in terms of the following four subproblems: ################## 1 This work was partly supported by ThomsonCSF/DSE and by the National Science Foundation under grant CCR9002428. 2. Research at Princeton University partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, grant NSFSTC8809648, and the Office of Naval Research, contract N0001487K0467.    2  P1. Decide whether a connected graph G is planar. P2. Find a minimal set of edges the removal of which will render the remaining part of G planar. P3. Gi...
Counting Embeddings of Planar Graphs Using DFS Trees
 SIAM Journal on Discrete Mathematics
, 1993
"... Previously counting embeddings of planar graphs [5] used PQ trees and was restricted to biconnected graphs. Although the PQ tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give for ..."
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Previously counting embeddings of planar graphs [5] used PQ trees and was restricted to biconnected graphs. Although the PQ tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give formulas that count the number of embeddings of general planar graphs (not necessarily connected or biconnected) in O (n) arithmetic steps, where n is the number of vertices of the input graph. Finally, our algorithm can be extended to generate all embeddings of a planar graph in linear time with respect to the output. Key words. graph, depth first search, embedding, planar graph, articulation point, connected component AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [14], Wu stated four basic planar graph problems: 1. Decide whether a connected graph G is planar. 2. Find a minimal set of edges the removal of which will render the remaining part of G planar. ...