Book contents
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
Obstructions for the Disk and the Cylinder Embedding Extension Problems
Published online by Cambridge University Press: 06 December 2010
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
Summary
Let S be a closed surface with boundary ∂S and let G be a graph. Let K ⊆ G be a subgraph embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S that coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the disk or the cylinder. Linear time algorithms are presented that either find an embedding extension, or return an obstruction to the existence of extensions. These results are to be used as the corner stones in the design of linear time algorithms for the embeddability of graphs in an arbitrary surface and for solving more general embedding extension problems.
Introduction
Let K be a subgraph of G. A K-component or a K-bridge in G is a subgraph of G that is either an edge e ∈ E(G)\E(K) (together with its endpoints) that has both endpoints in K, or it is a connected component of G − V(K) together with all edges (and their endpoints) between this component and K. Each edge of a K-component R having an endpoint in K is a foot of R. The vertices of R ∩ K are the vertices of attachment of R. A vertex of K of degree in K different from 2 is a main vertex of K.
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- Information
- Combinatorics, Geometry and ProbabilityA Tribute to Paul Erdös, pp. 493 - 524Publisher: Cambridge University PressPrint publication year: 1997