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By C.F.Picard

* Publisher: North Holland

* Number Of Pages: 446

* Publication Date: 1980-01-15

* ISBN-10 / ASIN: 0444852395

* ISBN-13 / EAN: 9780444852397

PREFACE

It gives me great pleasure to see the publication now of the

English version of my work on graphs and questionnaires. As

there are already a great many execllent texts on graph theory

in the English language, it has seemed appropriate to shift

the emphasis more firmly onto the subject of questionnaires by

reducing the graph theoretical content from the first volume of

the French edition to three chapters. There we set out those

elements of graph theory essential for the development of the

mathematical techniques used in the theory of questionnaires.

Chapter 3 is devoted to a discussion of several operations on

graphs which, although they have appeared in many separate

publications, have not yet been given a systematic treatment

in any other book. Chapter 1 corresponds to chapters 1, 2 and

3 of the French edition, chapter 2 to chapters 4 and 5 and

chapter 3 to chapter 6 and chapter 7. The reduction has been

achieved by leaving out certain sections for which no summary

is given. The remaining topics are carried over without

modification.

Chapters 4 to 10 constitute the translation of the second

volume of the French edition.

Following the bibliography arranged by chapter, a supplementary

list of papers on the subject of questionnaires rounds off the

work.

Except as indicated by the above remarks, the account of the

contents given in the Preface to the French edition naturally

still applies.

The topics dealt with in this book have aroused the interest of

several authors who have been able to make original contributions

to the theory and to guide young postgraduate students

into this line of research.

This translation is leaving the presses at just the left time

and I hope that a favourable reception by research workers,

engineers and technicians will facilitate further progress to

new extensions and applications.

I should like to thank the translators who have often contributed

appreciable improvements as compared with the French text

and have made every effort to spot any errors.

I am also grateful to the North-Holland Publishing Company for

the care they have taken over the preparation and presentation

of the book.

CONTENTS

Preface

Preface to the French edition

Chapter I Fundamental properties of graphs

Exercises

Ordered pairs and product sets

The graph concept

Elementary operations and transitive

closures

Connectivity, equivalence and preorder

Graph representations

Various definitions of graphs

Graph isomorphisms

Adjacency matrices

The incidence matrix of a graph

Computer representation of graphs

Valuations

Coding

Paths, circuits and cocircuits

Chains and concatenation

Cocircuits, cocycles and cycles

Chapter II Latticoids and arborescences

Exercises

Circuitless graphs

Arborescences and trees

Arborescences and data processing

Simplexes and arborescences

Monoids and arborescences

Chains, paths and arborescent

procedures

Coding

Finite and infinite graphs

Transportation networks

Chapter III Operations on graphs

Exercises

General definitions

Unary operations

Transformations

Cartesian operations

Product and sum

Classes of vertices

Connectivities

Valuations

Latticoid operations

Chapter IV General properties of questionnaires

Exercises

Preliminaries

The concept of a questionnaire

Axioms and definitions

Cutsets of a questionnaire

Partitions of the answers

Probabilities in an arborescent

questionnaire

Routing

The arborescence of paths in a latticoid

questionnaire

Compatible arborescent questionnaires

Probabilities in a latticoid

questionnaire

Probabilities of the vertices

Probabilities of the arcs and

conditioning

Example of a semantic

A restriction of the theory

Routing length

Chapter V The construction of questionnaires

Exercises

Operations on questionnaires

Definitions

Operations and routing length

Valuations on the answers and the arcs

L-optimal supports

Homogeneous questionnaires

a-I is a divisor of N-l

a-I is not a divisor of N-l

Heterogeneous questionnaires

Properties of arborescent questionnaires

The number of vertices and notation

Arborescences of minimal height

Questionnaires with balances support

Arborescences and questionnaires of

maximal height

Extremal properties of the supports

Chapter VI Optimal routing

Determination of an L-optimal

questionnaire

Necessary conditions for L-optimality

Substitutions of arcs

Transfers of arborescences

Sub-questionnaires

A sufficient condition for L-optimality

Huffman's algorithm

Questionnaires and coding

Equiprobable polychotomic questionnaires

Exercises

A characteristic property of homogeneous

balanced arborescences

Equiprobable dichotomic questionnaires

Optimal questionnaires

Routing in a dichotomic questionnaire

which is not optimal

Chapter VII Informational study of questionnaires

Exercises

Introduction to information

Hartley and Shannon's forms

Questionnaires in the sense of Shannon

Axiomatics of information

Faddeev's axioms

Some axiom systems

Properties of information

Convexity and concavity

Independence and dependence

Processed information and transmitted

information

Other definitions of information

Information for incomplete distributions

Measure, probability and information

Probability and information

Information for measure spaces

Non-probabilistic questionnaires

Chapter VIII Information and routing length

Information and routing in questionnaires

Inefficiency and noise

L-optimal questionnaires (LH ~ I)

Questionnaire product of two polychotomic

questionnaires

Heterogeneous questionnaires

Contribution of information

Maximization of processed information

and Shannon-Fano's algorithm

Partitions in equiprobable dichotomic

questionnaires and choice

Minimization of the contributed

information and Huffman's algorithm

Dichotomic questionnaires

Polychotomic questionnaires in the strict

sense

Heterogeneous questionnaires

Polychotomic questionnaires in the broad

sense

Informational interpretation of Huffman's

algorithm

Heterogeneous information and acquisition

Quasi-questionnaires

Quasi-answers and quasi-questions

Exercises

Instantaneous codes

Upper bounds for L-optimal questionnaires

Chapter IX Conditioning of the questions and answers

Limitations and extensions

Utilities of the answers

Useful length

Useful information

Cost of the questions

Costs and expenses

Free costs

Binding of the costs to the bases

Logarithmic costs

Questionnaires in the sense of Campbell

Questionnaires and Renyi's information

Charges and expenses

Questionnaires in the broad sense

Infinite questionnaires

Questionnaires with circuits

Flow charts and circuits

Realizable questionnaires

Constraints in questionnaires

A detection problem

Partitions and formable questions

Arborescent realizable L-optimal

questionnaires

The equivalence of constraints and costs

Questionnaires in practice

The dynamic aspect of interrogation

A random experiment

Absorption tests

Weighings

Interrogations, comparisons, sortings

Indirect interrogations and pseudoquestionnaires

Direct and indirect interrogations

Pseudoquestionnaires

Routing, information, convergence

Applications to pattern recognition

Diagnosis aid

Segmentation in a population

Word recognition

Comparisons and questions

Compatible realizable latticoids and

arborescences

Products of arborescent questionnaires

Sequential questionnaires

Questionnaires for sorting

L-optimal sorting

Realizable sortings

Problems

Solutions to Problems

Tables

Bibliography

Index

Main Symbols

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