Depth-First Search and Kuratowski Subgraphs

Williamson, S. Gill

doi:10.1145/1634.322451

Cited by 54 publications

(31 citation statements)

References 2 publications

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“…Description. Using a linear-time planarity algorithm that actually outputs an embedding, such as [13] or [18], we can assume that G is a plane graph. The algorithm is recursive.…”

Section: Proof the Graph Gmentioning

confidence: 99%

Three-coloring triangle-free planar graphs in linear time

Dvořák

Kawarabayashi

Thomas

2011

ACM Trans. Algorithms

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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.

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“…Description. Using a linear-time planarity algorithm that actually outputs an embedding, such as [13] or [18], we can assume that G is a plane graph. The algorithm is recursive.…”

Section: Proof the Graph Gmentioning

confidence: 99%

Three-coloring triangle-free planar graphs in linear time

Dvořák

Kawarabayashi

Thomas

2011

ACM Trans. Algorithms

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“…Checking whether a particular 3-cut is strong can be accomplished in O(n) time using depth-first search. All other operations, including checking if a graph is planar or has treewidth at most CH, can be performed in linear time [166,32].…”

Section: O(iv(g)i)mentioning

confidence: 99%

The Bidimensionality Theory and Its Algorithmic Applications

Demaine¹,

Hajiaghayi²

2007

The Computer Journal

184

130

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Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NPhard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k 2 ), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results. The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor.In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L 1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O( lo-gn) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be O('Iogn).We also prove various approximate max-flow/min-vertex-3 cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O(Vlo/g) pseudo-approximation for finding balanced vertex separators in general graphs. Furthermore, we obta...

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“…As Williamson [22] once said of Kuratowski subgraph isolation, it is desirable to have not one but several basically different optimal methods for solving a problem because the requirement of optimality forces the emergence of greater insight into underlying theoretic phenomena. An elegant case in point is found in the comparison of the first linear time K 4 search in [17] with the linear time K 4 search by Asano [1].…”

Section: Introductionmentioning

confidence: 99%

Subgraph Homeomorphism via the Edge Addition Planarity Algorithm

Boyer¹

2012

JGAA

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This paper extends the edge addition planarity algorithm from Boyer and Myrvold to provide a new way of solving the subgraph homeomorphism problem for K2,3, K4, and K3,3. These extensions derive much of their behavior and correctness from the edge addition planarity algorithm, providing an alternative perspective on these subgraph homeomorphism problems based on affinity with planarity rather than triconnectivity. Reference implementations of these algorithms have been made available in an open source project

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Depth-First Search and Kuratowski Subgraphs

Abstract: Lel G = (V, E) be a nonplanar graph. The method of using depth-first techniques to extract a Kuratowski .,;ubgraph in time O( I V[ ) is shown.

Cited by 54 publications

References 2 publications

Three-coloring triangle-free planar graphs in linear time

Three-coloring triangle-free planar graphs in linear time

The Bidimensionality Theory and Its Algorithmic Applications

Subgraph Homeomorphism via the Edge Addition Planarity Algorithm

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