A linear algorithm for embedding planar graphs using PQ-trees

Algorithmica, 1996

We give a detailed description of the embedding phase of the Hopcroft and Tarjan planarity testing algorithm. The embedding phase runs in linear time. An implementation based on this paper can be found in MMN93]. An undirected graph G = (V; E) is called planar if it can be mapped into the plane without edge crossings, i.e., the vertices of G are mapped into distinct points in the plane and the edges of G are mapped into disjoint Jordan curves connecting their endpoints. Such a mapping is called a planar geometric embedding. Two geometric planar embeddings are called equivalent if there is an homeomorphism of the plane transforming one into the other. An equivalence class of geometric planar embeddings is called a planar topological embedding or simply planar embedding. A planar embedding of a planar graph induces a cyclic ordering on the edges incident to any xed vertex, namely the clockwise ordering of the edges around their common endpoint. A graph G together with a cyclic ordering on the edges incident to each vertex is called a combinatorial embedding, it is called a planar combinatorial embedding if it is induced by some planar embedding. Di erent planar embeddings can give rise to the same combinatorial embedding. However, a planar combinatorial embedding of a connected graph uniquely determines its topological embedding on the sphere. In the plane, it determines the topological embedding up to selection of the outer face. Recall that an embedding into the plane can be obtained from an embedding on the sphere by polar projection. The pole can be put into any face. There are linear time algorithms dFPP91, Sch90] to convert a planar combinatorial embedding into a geometric embedding, e.g., the algorithm by Schnyder puts the vertices of an n node graph onto an (n 2) (n 2) grid and realizes the edges by straight-line segments. Hopcroft and Tarjan HT74] gave an algorithm that tests the planarity of an undirected graph in linear time. Alternative linear time algorithms were developed by Lempel, Even, and Cederbaum LEC67, ET76], Booth and Lueker BL76], and Fraysseix and Rosenstiehl FR82]. Chiba, Nishizeki, Abe and Ozawa CNAO85] have shown how to extend the algorithm of Booth and Lueker so as to also construct a planar combinatorial embedding. Hopcroft and Tarjan also stated but gave no details that their planarity testing algorithm can be extended to also construct a planar combinatorial embedding. The textbook of the rst author Meh84, vol. 2, page 112] attempts to give more details (in less than one page) but the presentation is incorrect. We conclude that there is no published correct presentation of the embedding phase of the Hopcroft and Tarjan algorithm.

Proceedings of the sixth annual ACM …, 1974

The isomorphism problem for graphs G 1 and G 2 is to determine if there exists a oneto-one mapping of the vertices of G 1 onto the vertices of G 2 such that two vertices of G 1 are adjacent if and only if their images in G 2 are adjacent.

ACM Transactions on Algorithms, 2015

We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "oncas" behaviour -obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1989

Abstmct-In this paper we present two O ( n * ) planarization algorithms-PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on Lempel, Even, and Cederbaum's planarity testing algorithm [9] and its implementation using PQ-trees [8]. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. This algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G,, of a nonplanar graph G, algorithm MAXIMAL-PLANARIZE constructs a maximal planar subgraph of G which contains G,,. This latter algorithm can also be used to maximally planarize a biconnected planar graph.

Lecture Notes in Computer Science, 2007

We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most 11 log 2 n + 43. As a consequence, a canonic form of such graphs is computable in AC 1 by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in AC 1 . * Supported by an Alexander von Humboldt fellowship.

Lecture Notes in Computer Science, 2020

A plane graph is rectilinear planar if it admits an embeddingpreserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal O(n) time for any plane series-parallel graph G with n vertices. If G is rectilinear planar, an embedding-preserving rectilinear planar drawing of G can be constructed in O(n) time. Our result is based on a characterization of rectilinear planar series-parallel graphs in terms of intervals of orthogonal spirality that their components can have, and it leads to an algorithm that can be easily implemented.

Algorithmica, 2014

A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.

2006

We present a linear time algorithm for testing clustered planarity of c-connected clustered graphs and for computing a clustered planar embedding for such graphs. Our algorithm uses a decomposition of the input graph based on SPQR-trees and is the first linear time algorithm for clustered planarity testing. We define a normal form of clustered embeddings and show that a clustered graph is clustered planar if and only if any of its normal form embeddings is clustered planar. We also give a combinatorial characterization of clustered planar embeddings and show how to test clustered planarity of a given embedding of a clustered graph.

In this paper we study two problems related to the drawing of level graphs, that is, T -LEVEL PLA-NARITY and CLUSTERED-LEVEL PLANARITY. We show that both problems are N P-complete in the general case and that they become polynomial-time solvable when restricted to proper instances.

Algorithmica, 2016

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph G(V, E) and a function γ : V → {1, 2,. .. , k} and asks whether a planar drawing of G exists such that each edge is monotone in the y-direction and, for any u, v ∈ V with γ(u) < γ(v), it holds y(u) < y(v). The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. We show that the problem is polynomial-time solvable if G has a fixed planar embedding.

2003

In this paper we give a new description of the planarity testing and embedding algorithm presented by Boyer and Myrvold [2], providing, in our opinion, new insights on the combinatorial foundations of the algorithm. Especially, we give a detailed illustration of a fundamental phase of the algorithm, called walk-up, which was only succinctly illustrated in [2]. Also, we present an implementation of the algorithm and extensively test its efficiency against the most popular implementations of planarity testing algorithms. Further, as a side effect of the test activity, we propose a general overview of the state of the art (restricted to efficiency issues) of the planarity testing and embedding field.

Lecture Notes in Computer Science, 2014

An HV-restricted planar graph G is a planar graph with vertex-degree at most four and such that each edge is labeled either H (horizontal) or V (vertical). The HV-rectilinear planarity testing problem asks whether G admits a planar drawing where every edge labeled V is drawn as a vertical segment and every edge labeled H is drawn as a horizontal segment. We prove that HV-rectilinear planarity testing is NP-complete even for graphs having vertex degree at most three, which solves an open problem posed by both Manuch et al. (GD 2010) and Durucher et al. (LATIN 2014). We also show that HV-rectilinear planarity can be tested in polynomial time for partial 2-trees of maximum degree four, which extends a previous result by Durucher et al. (LATIN 2014) about HV-restricted planarity testing of biconnected outerplanar graphs of maximum degree three. When the test is positive, our algorithm returns an orthogonal representation of G that satisfies the given Hand V-labels on the edges.

A graph is planar if and only if it can be drawn in the plane without crossings. 5 I give a detailed exposition of a simple and efficient, yet poorly known algorithm for 6 planarity testing and embedding based on the left-right characterization of planarity.

Journal of Computer and System Sciences, 1973

An algorithm for determining whether two triconnected planar graphs are isomorphic is presented. The asymptotic growth rate of the algorithm is bounded by a constant times ! V ! log I V [ where I V I is the number of vertices in the graphs.

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022

Consider property testing on bounded degree graphs and let ε > 0 denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on ε. Recent advances in testing minor-freeness have proven that all additive and monotone properties of planar graphs can be tested in poly(ε −1) queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in poly(ε −1) queries? Is there a uniform query complexity upper bound for all planar properties, and what is the "hardest" such property to test? We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in exp(O(ε −2)) queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph requires exp(Ω(ε −2)) queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adaptation of the Newman-Sohler analysis that tracks dependencies on ε more carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other. We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in exp(O(ε −1 log ε −1)) queries. Moreover, testing isomorphism to a fixed forest requires exp(Ω(ε −1)) queries.