Kuratowski Subgraphs Research Articles

A Graph Library for Isabelle

In contrast to other areas of mathematics such as calculus, number theory or probability theory, there is currently no standard library for graph theory for the Isabelle/HOL proof assistant. We present a formalization of directed graphs and essential related concepts. The library supports general infinite directed graphs (digraphs) with labeled and parallel arcs, but care has been taken not to complicate reasoning on more restricted classes of digraphs. We use this library to formalize a characterization of Euler Digraphs and to verify a method of checking Kuratowski subgraphs used in the LEDA library.

A new planarity test

Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in a plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite involved. Several other approaches have also been developed for simplifying the planarity test. In this paper, we developed a very simple linear time testing algorithm based only on a depth-first search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PC-trees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graph-reduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. The recognition and embedding are actually done simultaneously in our algorithm (Booth and Lueker, 1976). The implementation of our algorithm is quite straightforward.

Covering genus-reducing edges by Kuratowski subgraphs

If G is a graph that minimally does not embed in a nonorientable surface, then each edge of G is in a subdivision of either K3,3 or K5. However, there is an example of a graph that minimally does not embed in the torus and some edge is in no subdivision of either K3.3 or K5. © 1996 John Wiley & Sons, Inc.

Menus of kuratowski subgraphs

Motivated by heuristic embedding algorithms, this paper is concerned with the organization of potentially large lists of Kuratowski subgraphs of an arbitrary nonpianar graph. A graphical structure called a "nearly Hamiltonian" graph is defined. It is shown that lists of Kuralowski subgraphs can be lexicographically organized in such structures. It is shown that any nonpianar graph contains such structures and at least one such structure with a nonempty list of Kuratowski subgraphs can be located in linear time in ihe edges of the graph.

Depth-First Search and Kuratowski Subgraphs

Soit G=(V, E) un graphe non planaire. On montre la methode utilisant les techniques de 1ere profondeur pour extraire sur un sous-graphe de Kuratowsky en un temps O(IVI)

Crossing-critical edges and Kuratowski subgraphs of a graph

An edge e of a graph G is said to be crossing-critical if cr( G − e) < cr( G), where cr( G) denotes the crossing number of G on the plane. It is proved that any crossing-critical edge e of a graph G for which cr( G − e) ≤ 1 belongs to a subdivision of K 5 or K 3, 3, the Kuratowski subgraphs of G. Further, as regards crossing-critical edges e of G for which cr( G − e) ≥ 5, it is shown that the properties of “being a crossing-critical edge of G” and “being contained in a Kuratowski subgraph of G” are independent.

Edges and Kuratowski Subgraphs of Non-Planar Graphs

It is proved that any edge of a 4-connected non-planar graph G of order at least 6 lies in a subdivision of K3,3 in G. For any 3-connected non-planar graph G of order at least 6 we show that G contains at most four edges which belong to no subdivisions of K3,3 in G.

How to Draw a Graph

W E use the definitions of (11). However, in deference to some recent attempts to unify the terminology of graph theory we replace the term 'circuit' by 'polygon', and 'degree' by 'valency'. A graph G is 3-connected (nodally 3-connected) if it is simple and non-separable and satisfies the following condition; if G is the union of two proper subgraphs H and K such that HnK consists solely of two vertices u and v, then one of H and K is a link-graph (arc-graph) with ends u and v. It should be noted that the union of two proper subgraphs H and K of G can be the whole of G only if each of H and K includes at least one edge or vertex not belonging to the other. In this paper we are concerned mainly with nodally 3-connected graphs, but a specialization to 3-connected graphs is made in § 12. In § 3 we discuss conditions for a nodally 3-connected graph to be planar, and in § 5 we discuss conditions for the existence of Kuratowski subgraphs of a given graph. In §§ 6-9 we show how to obtain a convex representation of a nodally 3-connected graph, without Kuratowski subgraphs, by solving a set of linear equations. Some extensions of these results to general graphs, with a proof of Kuratowski's theorem, are given in §§ 10-11. In § 12 we discuss the representation in the plane of a pair of dual graphs, and in § 13 we draw attention to some unsolved problems.