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, depth-first-search, 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 Thomson-CSF/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 NSF-STC88-09648, and the Office of Naval Research, contract N00014-87-K-0467. -- -- - 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...

We show how to maintain centers and medians for a collection of dynamic trees where edges may be inserted and deleted and node and edge weights may be changed. All updates are supported in O(log n) time, where n is the size of the tree(s) involved in the update.

Previously counting embeddings of planar graphs [5] used P-Q trees and was restricted to biconnected graphs. Although the P-Q 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. ...

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We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQR-trees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determine the conditions in which the implemented algorithm outperforms other graph matchers, which do not impose topological restrictions on graphs. We report experiments with our planar graph matcher tested against McKay’s, Ullmann’s, and SUBDUE’s (a graph-based data mining system) graph matchers.