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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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