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## RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE

*Stefanie Gerke, Colin McDiarmid, Angelika Steger, and Andreas Weißl*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0006
- Subject:
- Mathematics, Probability / Statistics

This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 ... More

## RANDOM PLANAR GRAPHS AND THE NUMBER OF PLANAR GRAPHS

*Marc Noy*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0014
- Subject:
- Mathematics, Probability / Statistics

This chapter surveys recent progress on the enumeration of labelled planar graphs and on the distribution of several parameters in random planar graphs. It shows how to use generating functions to ... More

## DATA REDUCTION AND PROBLEM KERNELS

*Rolf Niedermeier*

### in Invitation to Fixed-Parameter Algorithms

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0007
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents one of the most basic and most important techniques for developing fixed-parameter algorithms: data reduction and problem kernelization. It starts with some basic definitions ... More

## TREE DECOMPOSITIONS OF GRAPHS

*Rolf Niedermeier*

### in Invitation to Fixed-Parameter Algorithms

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0010
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter provides an introduction to tree decomposition and treewidth, important concepts from modern graph theory. Treewidth is one of the best studied and most significant structural ... More

## DEPTH-BOUNDED SEARCH TREES

*Rolf Niedermeier*

### in Invitation to Fixed-Parameter Algorithms

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0008
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents the second very basic design technique for fixed-parameter algorithms: depth-bounded search trees. It starts with simple observations and some basic definitions and facts, ... More

## SELECTED CASE STUDIES

*Rolf Niedermeier*

### in Invitation to Fixed-Parameter Algorithms

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566076
- eISBN:
- 9780191713910
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566076.003.0015
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter briefly illustrates numerous case studies for parameterized complexity analysis. It begins with problems on planar and more general graphs, then discusses graph modification and other ... More

## CONNECTION MATRICES

*László Lovász*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0012
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces a matrix, the kth connection matrix, for a given graph parameter and integer k ≥ 0. The properties of these matrices are closely connected with properties of the parameter. ... More

## Drawing Graphs

*Arthur Benjamin, Gary Chartrand, and Ping Zhang*

### in The Fascinating World of Graph Theory

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175638
- eISBN:
- 9781400852000
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175638.003.0010
- Subject:
- Mathematics, Applied Mathematics

This chapter considers the concept of planar graph and its underlying theory. It begins with a discussion of the Three Houses and Three Utilities Problem and how it can be represented by a graph. It ... More

## Counting, Sampling, and Statistical Physics

*Cristopher Moore and Stephan Mertens*

### in The Nature of Computation

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199233212
- eISBN:
- 9780191775079
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233212.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. ... More

## Synchronizing Graphs

*Arthur Benjamin, Gary Chartrand, and Ping Zhang*

### in The Fascinating World of Graph Theory

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175638
- eISBN:
- 9781400852000
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175638.003.0012
- Subject:
- Mathematics, Applied Mathematics

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that ... More

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