Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Euler's theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.

We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only its topological ends are added as new points, while rays from other ends are made to converge to certain vertices.

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 adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts.

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. ...

We obtain three results concerning topological paths ands circles in the end compactification |G| of a locally finite connected graph G. Confirming a conjecture of Diestel we show that through every edge set E ∈ C there is a topological Euler tour, a continuous map from the circle S 1 to the end compactification |G | of G that traverses every edge in E exactly once and traverses no other edge. Second, we show that for every sequence (τi)i∈N of topological x–y paths in |G| there is a topological x–y path in |G | all of whose edges lie eventually in every member of some fixed subsequence of (τi). It is pointed out that this simple fact has several applications some of which reach out of the realm of |G|. Third, we show that every set of edges not containing a finite odd cut of G extends to an element of C.

Abstract. In this paper we develop a discrete, T0 topology in which (1) closed sets play a more prominent role than open sets, (2) atoms comprising the space have discrete dimension, which (3) is used to define boundary elements, and (4) configurations within the topology can have connectivity (or separation) of different degrees. To justify this discrete, closure based topological approach we use it to establish an n-dimensional Jordan surface theorem of some interest. As surfaces in digital imagery are increasingly rendered by triangulated decompositions, this kind of discrete topology can replace the highly regular pixel approach as an abstract model of n-dimensional computational geometry. 1 Axiomatic Basis Let U be a universe of arbitrary elements, or as we will call them, atoms. We let R denote a binary relation on U. We denote the identity relation I on U by R 0. Relational composition is defined in the usual way, so R k = R◦R k−1, and in particular, R 1 ◦R 0 = R◦I = R. Notationally, we denote elements (x, z) ∈ R k by x.R k.z. 1 Then, x.R k = {z | x.R k.z} and X.R k = {z |∃x ∈ X, x.R k.z}. In addition to R, we assume an integer function δ: U → Z that satisfies the following basic axiom x.R.z implies δ(x)>δ(z). (1) An easy induction on k establishes that x.R k.z also implies δ(x)>δ(z). Consequently, Lemma 1. If x.R m.z and z.R n.x then n = m =0and x = z. Proof. If x.Rm.z then δ(x)>δ(z), so if z.Rn.x we have δ(z)>δ(x), a contradiction unless m = n = 0 and x = z.

We extend Tutte's result that in a finite 3-connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends.

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