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.

The central topic of this thesis are criteria and tests which reveal whether a given clustered graph allows an embedding in the plane for which no edges and clusters intersect. Together with their definition in 1996, a notion of planarity was presented for clustered graphs, as well as an algorithm which tests this planarity for a given clustered graph in linear time. The algorithm however expects each cluster to be connected. For general clustered graphs, no efficient algorithm is yet known, neither is the computational complexity of the problem. This work presents algorithms which extend the class of clustered graphs for which planarity can be tested in polynomial time. A second part considers a weak form of planarity, and shows that a polynomial time test for this form also yields a polynomial time test for the classical definition. Furthermore, an attempt is made, by means of a characterization of the weak realizability problem in terms of forbidden subgraphs, to gain a similar characterization of the weak form of cluster planarity.

Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 1930, an efficient solution to it was found only in the seventies of the last century. Planar graphs play an important role both in the graph theory and in the graph drawing areas. In fact, planar graphs have several interesting properties: for example they are sparse, four-colorable, allow a number of operations to be performed efficiently, and their structure can be elegantly described by an SPQR-tree (see Section 3.1.2). From the information visualization perspective, instead, as edge crossings turn out to be the main culprit for reducing readability, planar drawings of graphs are considered clear and comprehensible. As a matter of fact, the study of planarity has motivated much of the development of graph theory. In this chapter we review the number of alternative algorithms available in the literature for efficiently testing planarity and computing planar embeddings. Some of these algorithms

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