Grötzsch’s theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm). We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grötzsch’s theorem. 1

Let K be a subgraph of G. It is shown that if G is 3–connected modulo K then it is possible to replace branches of K by other branches joining same pairs of main vertices of K such that G has no local bridges with respect to the new subgraph K. A linear time algorithm is presented that either performs such a task, or finds a Kuratowski subgraph K5 or K3,3 in a subgraph of G formed by a branch e and local bridges on e. This result is needed in linear time algorithms for embedding graphs in surfaces.

We describe an implementation of the Hopcroft and Tarjan planarity test and embedding algorithm. The program tests the planarity of the input graph and either constructs a combinatorial embedding (if the graph is planar) or exhibits a Kuratowski subgraph (if the graph is non-planar).

In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PC-trees (generalized from PQ-trees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not planar, and how to obtain the embedding alongside the testing algorithm. We have implemented the algorithm using LEDA and an object code is available at

A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...

I give a self-contained exposition of a linear-time planarity algorithm of Shih and Hsu.

I give a self-contained exposition of a linear-time planarity algorithm of Shih and Hsu. 20 December 1995 Revised 6 June 1997 Partially supported by NSF under Grant No. DMS-9303761 and by ONR under Grant No. N0001493 -1-0325. 1. INTRODUCTION There are four linear-time planarity algorithms that I am aware of [1, 3, 4, 5, 6], of which the one in [5] (thereafter referred to as the S&H algorithm) seems to be the simplest. The purpose of this note is to give a self-contained presentation of this algorithm. In section 2 we prove a structural result that is the basis for the correctness of the S&H algorithm, and illustrate it by deducing Kuratowski's theorem from it. The structural result immediately gives a quadratic planarity algorithm. In Section 3 we show how to implement it to run in linear time and linear space for 3--connected graphs on a RAM machine. 2. PLANAR GRAPHS By a graph we mean a finite, undirected, simple graph. This is without loss of generality, because a multigraph ...

An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1

Abstract — We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has a vertex set X of order at most k such that G − X is planar (we call this class of graphs k-apex), and if this is the case, computes a drawing of the graph in the plane after deleting at most k vertices. In fact, in this case, we shall determine the minimum value l ≤ k such that after deleting some l vertices, the resulting graph is planar. If this is not the case, then the algorithm gives rise to a minor which is not k-apex and is minimal with this property. This answers the question posed by Cabello and Mohar in 2005, and by Kawarabayashi and Reed (STOC’07), respectively. Note that the case k = 0 is the planarity case. Thus our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar in linear time. Our algorithm can be also compared to the algorithms by Mohar (STOC’96 and Siam J. Discrete Math 2001) for testing the embeddability of an input graph in a fixed surface in linear time, by Kawarabayashi and Mohar (STOC’08) for testing polyhedral embeddability of an input graph in a fixed surface in linear time, and by Kawarabayashi and Reed (STOC’07) for testing the fixed crossing number in linear time. Note that deciding the genus of k-apex graphs is NP-complete, even for k = 1, as shown by Mohar. Thus k-apex graphs are very different from bounded genus graphs in a sense. In addition, for any fixed c, k, we apply our algorithm to obtain a linear time approximation scheme for weighted TSP, and for minimum weighted c-edge-connected submultigraph, respectively, for k-apex graphs. (In this case, an embedding of a k-apex graph is not given in the input). The first result generalizes the recent planar result by Klein (FOCS’05), while the second result generalizes Czumaj et al. (SODA’04). We also extend several optimization results for planar graphs by Baker (J. ACM. 1994) and others to k-apex graphs.

Abstract. A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch-and-Cut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation. 1