Springer Monographs in Mathematics For further volumes: www.springer.com/series/3733 CuuDuongThanCong.com Jørgen Bang-Jensen r Gregory Z Gutin Digraphs Theory, Algorithms and Applications Second edition CuuDuongThanCong.com Prof Jørgen Bang-Jensen University of Southern Denmark Dept Mathematics & Computer Science Campusvej 55 5230 Odense Denmark jbj@imada.sdu.dk Prof Gregory Z Gutin Royal Holloway Univ London Dept Computer Science Egham Hill Egham, Surrey United Kingdom TW20 0EX gutin@cs.rhul.ac.uk ISSN 1439-7382 ISBN 978-1-84800-997-4 (hardcover) e-ISBN 978-1-84800-998-1 ISBN 978-0-85729-041-0 (softcover) DOI 10.1007/978-1-84800-998-1 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008939033 Mathematics Subject Classification (2000): 05C20, 05C38, 05C40, 05C45, 05C70, 05C85, 05C90, 05C99, 68R10, 68Q25, 68W05, 68W40, 90B06, 90B70, 90C35, 94C15 © Springer-Verlag London Limited 2001, 2009, First softcover printing 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency Enquiries concerning reproduction outside those terms should be sent to the publishers The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) CuuDuongThanCong.com To Lene and Irina CuuDuongThanCong.com Preface to the Second Edition The theory of graphs can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs) While there are many books on undirected graphs with new ones coming out regularly, the first edition of Digraphs, which was published in 2000, is the only modern book on graph theory covering more than a small fraction of the theory of directed graphs Since we wrote the first edition, the theory of directed graphs has continued to evolve at a high speed; many important results, including some of the conjectures from the first edition, have been proved and new methods were developed Hence a new completely revised version became necessary Instead of merely adding some new results and deleting a number of old ones, we took the opportunity to reorganize the book and increase the number of chapters from 12 to 18 This allows us to treat, in separate chapters, important topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, applications, as well as packing, covering and decompositions of digraphs We have added a large number of open problems to the second edition and the book now contains more than 150 open problems and conjectures, almost twice as many as the first edition In order to avoid the book becoming unacceptably long, we had to remove a significant portion of material from the first edition We have tried to this as carefully as possible so that most of the information in the first edition is still available, along with a very large number of new results Even though this book should not be seen as an encyclopedia on directed graphs, we included as many important results as possible The book contains a considerable number of proofs, illustrating various approaches and techniques used in digraph theory and algorithms One of the main features of this book is the strong emphasis on algorithms This is something which is regrettably omitted in many books on graphs Algorithms on (directed) graphs often play an important role in problems arising in several areas, including computer science and operations research Secondly, many problems on (directed) graphs are inherently algorithmic Hence, whenever possible we give constructive proofs of the results in the book From these proofs one can very often extract an efficient algorithm for the problem studied Even though we describe many algorithms, partly vii CuuDuongThanCong.com viii Preface due to space limitations, we not supply all the details necessary in order to implement these algorithms The latter (often highly nontrivial step) is a science in itself and we refer the reader to books on data structures and algorithms Another important feature is the large number of exercises which not only helps the reader to improve his or her understanding of the material, but also complements the results introduced in the text by covering even more material Attempting these exercises will help the reader to master the subject and its main techniques Through its broad coverage and the exercises, stretching from easy to quite difficult, the book will be useful for courses on subjects such as (di)graph theory, combinatorial optimization and graph algorithms Furthermore, it can be used for more focused courses on topics such as flows, cycles and connectivity The book contains a large number of illustrations This will help the reader to understand otherwise difficult concepts and proofs To facilitate the use of this book as a reference book and as a graduate textbook, we have added comprehensive symbol and subject indexes It is our hope that the organization of the book, as well as detailed subject index, will help many readers to find what they are looking for without having to read through whole chapters Due to our experience, we think that the book will be a useful teaching and reference resource for several decades to come Highlights We cover the majority of important topics on digraphs ranging from quite elementary to very advanced ones One of the main features of the second edition is the focus on open problems and the book contains more than 150 open problems or conjectures, thus making it a rich source for future research By organizing the book so as to single out important areas, we hope to make it easy for the readers to find results and problems of their interest Below we give a brief outline of some of the main highlights of this book Readers who are looking for more detailed information are advised to consult the list of contents or the subject index at the end of the book Chapter contains most of the terminology and notation used in this book as well as several basic results These are not only used frequently in other chapters, but also serve as illustrations of digraph concepts Chapter is devoted to describing several important classes of directed graphs, such as line digraphs, the de Bruijn and Kautz digraphs, digraphs of bounded tree-width, digraphs of bounded directed widths, planar digraphs and generalizations of tournaments We concentrate on characterization, recognition and decomposition of these classes Many properties of these classes are studied in more detail in the rest of the book Chapters and cover distances and flows in networks Although the basic concepts of these two topics are elementary, both theoretical and al- CuuDuongThanCong.com Preface ix gorithmic aspects of distances in digraphs as well as flows in networks are of great importance, due to their high applicability to other problems on digraphs and large number of practical applications, in particular, as a powerful modelling tool The main part of Chapter is devoted to minimization and maximization of distance parameters in digraphs In the self-contained Chapter 4, which may be used for a course on flows in networks, we cover basic topics on flows in networks, including a number of important applications to other (di)graph problems Although there are several comprehensive books on flows, we believe that our fairly short and yet quite detailed account of the topic will give the majority of readers sufficient knowledge of the area The reader who masters the techniques described in this chapter will be well equipped for solving many problems arising in practice Connectivity in (di)graphs is a very important topic It contains numerous deep and beautiful results and has applications to other areas of graph theory and mathematics in general It has various applications to other areas of research as well We give a comprehensive account of connectivity topics and devote Chapters 5, 10, 12, 14 and parts of Chapter 11 to different aspects of connectivity Chapter contains basic topics such as ear-decompositions, Menger’s theorem, algorithms for determining the connectivity of a digraph as well as advanced topics such as properties of minimally k-(arc)-strong digraphs and critically k-strong digraphs Chapter 10 deals with problems concerning (arc-)disjoint linkings with prescribed initial and terminal vertices in digraphs We prove that the 2linkage problem is N P-complete for arbitrary digraphs, but polynomially solvable for acyclic digraphs Results on linkings in planar digraphs, eulerian digraphs as well as several generalizations of tournaments are discussed In Chapter 12 we study the problem of finding, in a k-(arc)-strong digraph, a small set of arcs (called a certificate) so that these arcs alone show that the digraph has the claimed connectivity These problems are generally N P-hard, so we give various approximation algorithms as well as polynomial algorithms for special classes of digraphs We illustrate an application due to Cheriyan and Thurimella of Mader’s results on minimally k-(arc)-strong digraphs to the problem of finding a small certificate for k-(arc)-strong connectivity Finally, we also discuss recent results due to Gabow et al on directed multigraphs In Chapter 14 we describe the splitting technique due to Mader and Lov´ asz and illustrate its usefulness by giving an algorithm, due to Frank, for finding a minimum cardinality set of new arcs whose addition to a digraph D increases its arc-strong connectivity to a prescribed number We also discuss a number of results related to increasing the connectivity by reversing arcs In Chapter 11 the famous theorem by Nash-Williams on orientations preserving a high degree of local arc-strong connectivity is described and the CuuDuongThanCong.com x Preface weak version dealing with uniform arc-strong connectivities is proved using splitting techniques We also discuss extensions of these results, including recent ones by Kir´ aly and Szigeti We give a proof a Jord´ an’s result that every 18-connected graph has a 2-strong orientation Submodular flows form a powerful generalization of circulations in networks We introduce submodular flows and illustrate how to use this tool to obtain (algorithmic) proofs of many important results in graph theory Finally we describe in detail an application, due to Frank, of submodular flows to the problem of orienting a mixed graph in order to maintain a prescribed degree of arc-strong connectivity In Chapter we give a detailed account of results concerning the existence of hamiltonian paths and cycles in digraphs Many results of this chapter deal with generalizations of tournaments The reader will see that several of these much larger classes of digraphs share various nice properties with tournaments In particular the hamiltonian path and cycle problems are polynomially solvable for most of these classes The chapter illustrates various methods (such as the multi-insertion technique) for proving hamiltonicity In Chapter we describe a number of interesting topics related to restricted hamiltonian paths and cycles These include hamiltonian paths with prescribed end-vertices and orientations of hamiltonian paths and cycles in tournaments We cover one of the main ingredients in the proof by Havet and Thomass´e of Rosenfeld’s conjecture on orientations of hamiltonian paths in tournaments Chapter describes results on (generally) non-hamiltonian cycles in digraphs We cover pancyclicity and the colour-coding technique by Alon, Yuster and Zwick and its application to yield polynomial algorithms for finding paths and cycles of ‘logarithmic’ length We discuss the even cycle problem, including Thomassen’s even cycle theorem We also cover short cycles in ´ am’s multipartite tournaments, the girth of a digraph, chords of cycles and Ad´ conjecture The chapter features various proof techniques including several algebraic, algorithmic, combinatorial and probabilistic methods Chapter is devoted to branchings, a very important structure generalizing spanning trees in graphs Branchings are not only of interest by themselves, they also play an important role in many proofs on digraphs We prove Tutte’s Matrix-Tree theorem on the number of distinct out-branchings in a digraph We give a number of recent results on branchings with bounds on the degrees or extremal number of leaves Edmonds’ theorem on arc-disjoint branchings is proved and several applications of this important result are described The problem of finding a minimum cost out-branching in a weighted digraph generalizes the minimum spanning tree problem We describe algorithms for finding such a branching Chapter 13 covers a number of very important results related to packing, covering and decompositions of digraphs We prove the Lucchesi-Younger theorem on arc-disjoint directed cuts, and give a number of results on arc- CuuDuongThanCong.com Preface xi disjoint hamiltonian paths and cycles We discuss results on decomposing highly connected tournaments into many spanning strong subdigraphs, including a conjecture which generalizes the famous Kelly conjecture We give a number of results on cycle factors with a prescribed number of cycles Finally, we give a full proof, due to Bessy and Thomass´e, of Gallai’s conjecture on the minimum number of cycles needed to cover all vertices in a strong digraph Chapter 15 deals with another very important topic, namely, how to destroy all cycles in a digraph by removing as few vertices or arcs as possible One of the main results in the chapter is that the feedback arc set problem is N P-hard already for tournaments This was conjectured by Bang-Jensen and Thomassen in 1992, but was proved only recently by three different sets of authors Another major result is the proof by Chen, Liu, Lu, O’Sullivan and Razgon and that the feedback arc set problem and the feedback vertex set problems are both fixed-parameter tractable We also include a scheme of a solution of Younger’s conjecture, by Reed, Robertson, Seymour and Thomas, on the relation between the number of disjoint cycles and the size of a minimum feedback vertex set in a digraph Chapter 16 deals with edge-coloured graphs, a topic which has a strong relation to directed graphs Alternating cycles in 2-edge-coloured graphs generalize the concept of cycles in digraphs Certain results on cycles in bipartite digraphs, such as the characterization of hamiltonian bipartite tournaments, are special cases of results for edge-coloured complete graphs There are useful implications in the other direction as well In particular, using results on hamiltonian cycles in bipartite tournaments, we prove a characterization of those 2-edge-coloured complete graphs which have an alternating hamiltonian cycle We briefly describe one important recent addition to the topic, i.e., a characterization by Feng, Giesen, Guo, Gutin, Jensen and Rafiey of edge-coloured complete multigraphs containing properly coloured Hamilton paths This characterization was conjectured in the first edition Chapter 17 describes a number of different applications of directed and edge-coloured graphs Through the topics treated we illustrate the diversity of the applications including some in quantum mechanics, bioinformatics, embedded computing and the traveling salesman problem Chapter 18 is included to make the book self-contained, by giving the reader a collection of the most relevant definitions and methods on algorithms and related areas Technical Remarks We have tried to rank exercises according to their expected difficulty Marks range from (−) to (++) in order of increasing difficulty The majority of exercises have no mark, indicating that they are of moderate difficulty An exercise marked (−) requires not much more than the understanding of the CuuDuongThanCong.com xii Preface main definitions and results A (+) exercise requires a non-trivial idea, or involves substantial work Finally, the few exercises which are ranked (++) require several deep ideas Inevitably, this labelling is subjective and some readers may not agree with this ranking in certain cases Some exercises have a header in boldface, which means that they cover an important or useful result not discussed in the text in detail We use the symbol to denote the end of a proof, or to indicate that either no proof will be given or the assertion is left as an exercise A few sections of the book require some basic knowledge of linear programming, while a few others require basic knowledge of probability theory We would be grateful to receive comments on the book They may be sent to us by email to jbj@imada.sdu.dk or gutin@cs.rhul.ac.uk We plan to have a web page containing information about misprints and other information about the book; see http://www.cs.rhul.ac.uk/books/dbook/ Acknowledgments We wish to thank the following colleagues for helpful assistance and suggestions regarding various versions of the first edition: Noga Alon, Alex Berg, Thomas Băohme, Adrian Bondy, Jens Clausen, Samvel Darbinyan, Charles Delorme, Reinhard Diestel, Odile Favaron, Herbert Fleischner, Andr´ as Frank, Yubao Guo, Vladimir Gurvich, Fr´ed´eric Havet, Jing Huang, Alice Hubenko, Bill Jackson, Tommy Jensen, Thor Johnson, Tibor Jord´ an, Ilia Krasikov, Hao Li, Martin Loebl, Gary MacGillivray, Wolfgang Mader, Crispin NashWilliams, Jarik Neˇsetˇril, Steven Noble, Erich Prisner, Gert Sabidussi, Lex Schrijver, Paul Seymour, Eng Guan Tay, Meike Tewes, St´ephan Thomass´e, Carsten Thomassen, Bjarne Toft, Lutz Volkmann, Anders Yeo and Ke-Min Zhang For help on the second edition we wish to thank the following colleagues: Stephanne Bessy, Paulo Feofiloff, Fr´ed´eric Havet, Tibor Jord´ an, Eun Jung Kim, Paul Medvedev, Morten Hegner Nielsen, Anders Sune Pedersen, Zolt´ an Szigeti and Anders Yeo We are very grateful to Karen Borthwick of Springer-Verlag, London for her help and encouragement and to David Bahr who did excellent work copyediting the book We also thank the anonymous reviewers used by Springer for providing us with encouragement and very useful feedback Last, but most importantly, we wish to thank our families, in particular our wives Lene and Irina, without whose constant support we would never have succeeded in completing this project Odense, Denmark London, UK September 2008 CuuDuongThanCong.com Jørgen Bang-Jensen Gregory Gutin Subject Index k-dangerous set, 475 data compression, 369 data dependency graph (DDG), 649 de Bruijn digraph, 44–47, 316 de-randomizing, 323 decision problem, 700 decomposable digraph, see also quasi-transitive digraph, 9, 220, 277 decomposition Φ-decomposition of a digraph, 10 into acyclic digraphs, 548 into arc-disjoint hamiltonian cycles, 515, 516 into strong spanning subdigraphs, 536–542 into strong subdigraphs, 548 of a graph into cliques, 424 of the arc set of regular tournaments, 516 decreasing subsequence, 519 deficiency of a one-way pair, 563, 564 k-degenerate graph, 433 degree of a vertex jth degree, 609 in a digraph, in a graph, 20 degree-constrained digraphs, 504 hamiltonian cycles, 233–243 deletion of a subdigraph, of arcs from a digraph, of multiple arcs, of vertices from a digraph, demand arc, 402 demand directed multigraph, 403 density of a digraph, 331 deorienting an arc, 574 dependent set of a matroid, 712 of an independence system, 657 depth-first search, 26–29 descendant in a DFS tree, 27 design, 645 deterministic finite automaton, 687 DFS, see also depth-first search, 195 DFS forest, 27 DFS tree, 27 backward arc, 27 cross arc, 27 descendant of a vertex in, 27 CuuDuongThanCong.com 781 forward arc, 27 root of, 27 DHM-construction, 630 diameter, 89, 100–115 minimizing, 101 minimum in orientation, 103 Moore bound on number of vertices, 101 versus degree, 44 dicut, 505 arc-disjoint, 506 crossing dicuts, 507 Woodall’s conjecture, 511 difference between two sets, digraph, corresponding to instance of 2SAT, 667 Dijkstra’s algorithm, 94–97, 123 dijoin, 505 disjoint dijoins, 511 Dilworth’s theorem, 521 3-dimensional matching problem, 551 Dinic’s algorithm, 148 for simple networks, 156 on unit capacity networks, 154 Dinitz conjecture, proof using kernels, 679–683 directed cactus, 501 cut, see dicut dual of a planar digraph, 590, 605 graph, see also digraph, multigraph, pseudograph, 4, 513 associated with a Markov chain, 677 directed path decomposition, 78 directed path-width, 78 directed Steiner problem with connectivity constraints, 499 directed tree-width, 79 disjoint cycles, 415, 513, 550, 596–600 versus feedback sets, 596 disjoint paths, 374 disjoint sets, distance from a set to another, 89 from a vertex to another, 88 distance classes from a vertex, 93 distances acyclic digraphs, 94 algorithms for finding, 91–100 782 Subject Index Bellman-Ford-Moore algorithm, 97–98 Dijkstra’s algorithm, 94–97 in complete biorientations, 97 dominated, dominated pair of vertices, 233 dominates, dominating pair of vertices, 233 drop algorithm, 497 dual of a matroid, 713 dynamic programming, 322 ear decomposition, 198, 481, 678 application of, 200, 487 ear of, 198 linear algorithm for, 200 edge of an undirected graph, 18 edge-coloured multigraph, 608, 625– 628 2-edge-coloured complete multigraph, 628–634 edge-colouring, 438 k-edge-connected, 20, 442 edge-connectivity, 206 algorithm to determine, 206 maximum adjacency ordering, 206 edge-cover, 483 edge-disjoint 2-linkage problem, 406 mixed branchings, 365 paths, 406 spanning trees, 348 trees, 478 Edmonds’ branching theorem, 205, 208, 346–350, 370, 372, 399 generalization of, 347 Edmonds-Giles theorem, 454 electronic circuit design, 583 element of a directed pseudograph, elementary operation, 696 ellipsoid method, 457, 565 embedding of a planar (di)graph in the plane, 72 end-vertex of a walk, 12 of an arc, entering arc, Euler trail, see eulerian trail properly coloured, 610 Euler’s formula, 73 Euler’s theorem, 23 eulerian CuuDuongThanCong.com (multi)graph, 441, 443 digraph, 102 directed multigraph, 13, 23, 401–407 orientation of a mixed graph, 452 oriented graph, 103 subgraph, 438 trail, 13, 718 weak linkage problem, 403 eulerian directed multigraph, see also regular digraph, 166, 174, 175, 180, 254, 416, 513, 557 decomposition into cycles, 180 Evans Conjecture, 537 even cycle, 12, 35, 428 in a k-regular digraph, 328 oriented graphs with many arcs, 430 even cycle problem, 324 even digraph, 326 even pancyclic, 336 even vertex-pancyclic digraph, 336 exponential-time algorithm, 705 extended Φ-digraph, 10 extended locally in-semicomplete digraph, 35, 86, 407, 409 extended locally out-semicomplete digraph, 276 extended locally semicomplete digraph, 35, 69, 276, 277 extended semicomplete digraph, 35, 52, 69, 246, 251, 252, 276, 280, 288, 309, 310, 336, 393, 481, 484–487, 501, 503, 521 hamiltonian cycle, 246 longest cycle, 246 MSSS problem, 484, 485 extended tournament, 275, 279–281, 288, 470 hamiltonian [x, y]-path, 280 proof using the structure of, 275 weakly hamiltonian-connected, 281 extension of a digraph, 10, 111, 393 of a graph, 21 extension-closed class of digraphs, 10, 36 face of a plane (di)graph, 72 facial cycle, 384, 590 factor of a digraph, family Subject Index 2-covering, 645 cross-free, 192, 209 crossing, 192 has an SDR, 645 intersecting, 192 laminar, 192, 209 mediated, 645 of sets, symmetric, 645 fan-in, fan-out in eulerian directed multigraphs, 416 feasibility theorem for circulations, 157 for crossing submodular flows, 458 for flows, 158 for fully submodular flows, 455 feasible k-commodity flow, 413 flow, 129 with balance vectors within intervals, 185 pairing, 444 submodular flow, 454–458 feedback arc set, 511, 583, 593, 606, 708 feedback arc set problem, 587, 707– 711 approximation algorithm, 593, 606 planar digraph, 590 feedback sets, 583–600 versus (arc)-disjoint cycles, 596 feedback vertex set, 583, 705 feedback vertex set problem, 587 Fibonacci heap, 96 finite automaton, 687 fixed-parameter algorithmics, 703 fixed-parameter tractable (FPT), 704 flow, 128 across a cut, 140 adding a residual flow, 138 application, see application of flows arc sum of two flows, 136 augmenting path, 141 balance vector of, 128 circulation, 133 cost of, 129 cycle flow, 136 decomposition, 136, 140, 180 demand of a cut, 159 CuuDuongThanCong.com 783 difference between two flows, 139 feasibility theorem, 158 feasible, 129, 156, 185, 449 integer, 128 maximal, 144, 148 maximum, see maximum flow problem maximum capacity augmenting path method, 185 netto flow, 129 optimal, 162 path flow, 136 residual network with respect to, 130 (s, t)-flow, 132 (s, t)-cut, 140 minimum value, 158 reducing general flows to, 132 relation to arc-strong connectivity in directed multigraphs, 194 value of, 132 Floyd-Warshall algorithm, 37, 99 Ford-Fulkerson algorithm, 142, 205 on real valued instances, 181 forefather, 196 forest, 21 forward arc for a path, 361 on an augmenting path, 141 with respect to an ordering, 530, 583, 592 fragment, 218 Frank’s orientation theorem, 452, 465 Frank-Jord´ an vertex-connectivity augmentation theorem, 565 free matroid, 713 fully G-supermodular function, 452 fully submodular function, 454 gadget for N P-completeness proof, 228, 375, 551 Gallai-Milgram theorem, 519 Gallai-Roy-Vitaver theorem, 432 game theory, 119 gap of a C-bypass, 241 Gaussian elimination, 674 generalized de Bruijn digraph, 47 generalized matching, 188 generating pair, 82 genetics, 670 784 Subject Index geometric random variable, 321 girth, 12, 125, 312, 329–332 global irregularity, 255 good cycle factor, 249–253 theorem, 250 good vertex with respect to a locally optimal ordering, 602 gossip problem, 690 Gră otzsch graph, 473 graph, see also undirected graph, 18 graph Steiner problem, 366 greedy algorithm, 371 for independence systems, 657, 714 for matroids, 719 group flow, 436 half-duplex gossip problem, 691 Hall’s theorem, 173 Hamilton cycle, see hamiltonian cycle Hamilton Cycle Problem, 39, 700 Hamilton path, see hamiltonian path Hamilton walk, see hamiltonian walk hamiltonian (x, y)-path, 286, 288 hamiltonian [x, y]-path, 277, 280, 283, 285 hamiltonian connected, 286–289 hamiltonian cycle, 13, 37, 60, 207, 228, 230, 231, 257, 258, 268, 289–296, 303–305, 308–311, 315, 318, 319, 337, 480, 483, 485, 515–518, 529, 552, 662 alternating in 2-edge-coloured multigraph, 608 arc-disjoint hamiltonian cycles, 515, 536 avoiding prescribed arcs, 292– 296 containing prescribed arcs, 290, 292 in almost acyclic digraph, 304 in almost semicomplete digraph, 290 in undirected graph, 238 multipartite tournament, 293 power of a hamiltonian cycle, 517 properly coloured, 621, 635–640 quasi-transitive digraph, 256– 259, 485 semicomplete multipartite digraph, 244–256 CuuDuongThanCong.com sufficient conditions in terms of degrees, 233–243 hamiltonian digraph, 13 hamiltonian path, 13, 62, 91, 228, 231, 232, 245, 257, 275–289, 304, 313, 515–518, 530 alternating in 2-edge-coloured multigraph, 608 between two prescribed vertices, 277 in a tournament, 14, 697 in semicomplete bipartite digraph, 86 one end vertex prescribed, 275– 277 oriented, 297 properly coloured, 635 R´edei’s theorem, 14 hamiltonian walk, 13 Havet-Thomass´e theorem, 298 k-HCA problem, 290–292 head of a one-way pair, 563 of an arc, height function with respect to a preflow, 150, 182 hereditary set of digraphs, 69 heuristics, 660, 707 domination number, 661 domination ratio, 661 for N P-hard problems, 660, 707–711 Hoffman’s circulation theorem, 157, 457 hypergraph, 26, 104, 328 2-colourable, 26 2-colouring of, 26 edge of, 26 order of, 26 rank of, 26 transversal of edges, 364 uniform, 26 vertex of, 26 hypertournament, 607 implication class, 420, 423, 424 in-branching, see also out-branching, 22, 232 minimum cost, 497 k-in-critical set, 208, 209 in-degree of a vertex, in-generator of a digraph, 299 in-neighbour, in-neighbourhood, Subject Index pth in-neighbourhood, 88 in-path, 298 in-path-mergeable digraph, 58, 86 in-pseudodegree of a vertex, in-radius, 89 in-singular vertex with respect to a cycle, 247 in-tree, 22 incident to an arc, incomparable elements with respect to a partial order, 521 increasing subsequence, 519 independence number, 21, 85, 254, 519 effect on cycle factors, 530 independence oracle for a matroid, 343, 715 independence system, 657 base, 657 dependent set, 657 independent set, 657 uniform, 657 independent arcs (edges), 21 independent set, 427, 519, 718 of a matroid, 712 of an independence system, 657 independent set problem, 718 independent vertices, 21 index of a pair of alternating trails, 612 index-bounded weighting, 535 induced subdigraph, initial strong component, 17 initial vertex of a walk, 12 inserting one path into another, 239 instance of a problem, 700 integer multicommodity flow problem, 413 integrality theorem for maximum flows, 144 Intel Δ-prototype, 691 intercyclic digraph, 598 intermediate strong component, 17 internally disjoint paths, 13, 201, 224, 374 intersecting G-supermodular function, 452 family, 192 pair, 454 submodular function, 454 intersection digraph, 82 graph, 624 CuuDuongThanCong.com 785 number of a digraph, 82, 86 of digraphs, 38 interval digraph, 83 interval graph, 83, 115 interval of an oriented path, 298 length, 298 2-irreducible instance of k-ST problem, 405 irreducible alternating cycle subgraph, 632 irreducible cycle factor, 522 isomorphic directed pseudographs, graphs, 20 isomorphism, iterated line digraph, 43, 44 iterative compression, 594 iterative rounding of an LP solution, 500 Jordan curve theorem, 395 Kautz digraph, 46 Kelly’s conjecture, 516 kernel, 119–122 (k, l)-kernel, 119 kernel-imperfect digraph, 120 kernel-perfect digraph, 119, 680 kernel-solvable graph, 121 king, 115119, 126 Kircho matrix, 340 Kă onig’s theorem, 172 Kruskal’s algorithm, 342 Kuratowski’s theorem, 72 labelled digraph, labelling algorithm for maximum flow, 143 laminar family, 192, 209 Landau’s theorem, 449, 476 large packet radio network, 44 Las Vegas algorithm, 205 Latin square, 679 layered network, 146 leaving arc, length of a cycle, 12 of a path, 12 of a walk, 12 lexicographic 2-colouring, 421, 423 lexicographically smaller vertex, 421 line digraph, 39–44, 119, 316 iterated, 43 obstructions for, 42 786 Subject Index recognition, 41 linear ordering problem, 592 linear programming, 123, 145, 454, 457, 496, 500, 509, 535 k-linkage, 373, 598 k-linkage problem, 304, 374– 395, 398, 414 k-linked digraph, 373, 375, 379–398, 415 linking principle, 450, 461 list chromatic index, 682 list colouring, 680 list edge-colouring, 120, 679–683 literal, 667, 702 local arc-strong connectivity, 192 local edge-connectivity, 443 local in-tournament, see locally intournament digraph local irregularity, 255 local tournament, see locally tournament digraph local vertex-strong connectivity, 192 locally in-semicomplete digraph, see also locally outsemicomplete digraph, 57– 59, 85, 86, 231–233, 271, 273, 315, 426, 427 strong decomposition, 59 structure of non-strong, 59 locally in-tournament digraph, 57 locally optimal ordering, 601 locally optimal solution of an optimization problem, 708 locally out-semicomplete digraph, see also locally insemicomplete digraph locally semicomplete digraph, 57, 59– 68, 111, 221, 225, 233, 283– 285, 288, 303, 312–315, 336, 356, 357, 374, 387–389, 415, 421, 422, 425, 526 classification theorem, 68 complementary cycles, 526 extended, 276 generalization, 235 hamiltonian (x, y)-path, 288 hamiltonian [x, y]-path, 283, 285 hamiltonian connected, 288 independence number, 85 minimal separating set in, 225 non-round decomposable, 67 orientation of, 468 round decomposable, 62 CuuDuongThanCong.com semicomplete decomposition, 63 structure of non-strong, 61 weakly hamiltonian-connected, 285 locally tournament digraph, see also locally semicomplete digraph, 57, 289, 312, 313, 315, 319, 390, 422, 423, 425, 426, 468, 472, 473 characterization through orientations, 425 round, 61 longest (x, y)-path problem, 287 [x, y]-path problem, 288 alternating cycle, 627, 629 cycle extended semicomplete digraph, 484, 486 relation to chromatic number, 434 cycle problem, 53 path, 232 relation to chromatic number, 432 path problem, 53 acyclic digraph, 94, 123 loop, Lov´ asz’s local lemma, 327 Lov´ asz’s splitting theorem, 442, 475 lower bound on an arc, 127 removing from a network, 131 Lucchesi-Younger theorem, 506, 509 Mader’s directed splitting theorem, 555 main (n1 , , np )-blocks, 674 Markov chain, 677 marriage theorem, 174, 451, 476 matching, 21, 223, 443 perfect, 21, 427 matching diagram digraph, 83 matrix multiplication, 37 Matrix-tree theorem, 339 matroid, 343, 711–717, 719 matroid intersection problem, 343, 367, 461, 716, 717, 720 matroid partition problem, 715, 720 MAX-2-SAT, 670, 703, 718 Max-Flow Min-Cut theorem, 141, 172 relation to Menger’s theorem, 202 1-maximal cycle, 719 Subject Index maximal flow, 144, 148 maximal with respect to property P, maximum k-path subdigraph, 543 acyclic subdigraph problem, 592 adjacency ordering, 206 in-degree of a digraph, matching in bipartite graphs, 170, 171 monochromatic degree, 609 out-degree of a digraph, semi-degree of a digraph, with respect to property P, maximum flow algorithms, 142–156 capacity scaling algorithm, 183 Dinic’s algorithm, 148 for unit capacity networks, 154 Ford-Fulkerson algorithm, 142 maximum capacity augmenting path method, 186 MKM algorithm, 182 on simple networks, 156 push-relabel algorithm, 150 shortest augmenting paths, 147 maximum flow problem, 140–156 and arc-strong connectivity, 204 in unit capacity networks, 154 integrality theorem, 144 re-optimizing after small perturbation, 182 mean cost of a cycle, 165 mediated digraph, 644 mediation number, 644 member of a family of digraphs, of a family of sets, 192 Menger’s theorem, 201–206, 212, 224, 225, 287, 345, 347, 348, 370, 409, 410, 415, 443, 459, 469, 529, 557, 562, 580, 606 applied to sets of vertices, 225 refinement of, 224 relation to the Max-Flow MinCut theorem, 224 Mergesort, 699 merging paths in a digraph, see pathmergeable digraph meta-heuristics, 707, 709 Meyniel set, 236 Min-Flow Max-Demand theorem, 159 minimal CuuDuongThanCong.com 787 (x, y)-path, 53 vertex series-parallel digraphs, 48 minimally k-arc-strong directed multigraph, 207–213 k-edge-connected multigraph, 442, 475 k-strong digraph, 213–217 minimizing a submodular function, 457, 459, 478 minimum covering out-tree problem, 366 cycle factor problem, 521 diameter orientation, 103–115 diameter versus degree, 44 dijoin, 510 equivalent subdigraph, 39, 480 flow, 158 in-degree of a digraph, monochromatic degree, 609 out-degree of a digraph, path factor problem, 520 semi-degree of a digraph, spanning tree, 342, 371 minimum cost branching problem, 342 cover of directed cuts, 510 submodular flows, 458, 478, 509 minimum cost flows, 160–170 application to Chinese postman problem, 174 applied to a branching problem, 224 assignment problem, 169 buildup algorithm, 167 buildup theorem, 166 characterization, 163 cycle canceling algorithm, 163 integrality theorem, 165 strongly polynomial algorithm, 165 transportation problem, 169 minimum equivalent subdigraph, see MSSS problem minimum spanning strong subgraph problem, see MSSS problem mixed (multi)graph, 24, 225, 365, 451 mixed branchings, 365 mixed Chinese postman problem, 175 mixed graph arc of, 24 788 Subject Index biorientation of, 24 branchings, 365 bridge of, 24 complete biorientation of, 24 connected, 24 edge of, 24 orientation of, 24, 461–466 strong, 24 modular function, 448 monochromatic complete subgraph, 598 Monte Carlo algorithm, 205 Moon’s theorem, 16 Moore bound, 101 MSSS problem, 480–489 multi-insertion technique, 239 multicommodity flow, 413 multigraph, 18 multipartite digraph, 34–36 multipartite tournament, see also semicomplete multipartite digraph, 35, 112, 116, 117, 126, 247, 267, 272, 282, 293, 315, 316, 333, 526 multiple arcs, multiset, Nash-Williams’ orientation theorem, 442, 443, 459, 461, 475 negation, 667, 702 negative cycle, 88 detection, 98 effect on shortest path problems, 91 in residual network, 163 neighbour, neighbourhood, 4, 20 neighbourhood of a solution, 708 neighbouring solutions, 708 nested interval graph, 428 network, 127 augmenting path in, 141 balance vector of, 128 balanced vertex in, 129 capacity of arcs, 127 circulation in, 133 cost of arcs, 127 flow in, 128 layered, 146 lower bound on arcs, 127 maximum flow in, 140 residual with respect to a flow, 130 CuuDuongThanCong.com simple, 155 sink vertex in, 129 source vertex in, 129 unit capacity, 153 with bounds/costs on vertices, 134 network design, 44 network representation, 194, 202 nice tree decomposition, 75 non-deterministic finite automaton, 688 non-trivial λ-cut, 537 normal biorientation, 121 nowhere-zero flow Γ -flow, 436 k-flow, 435–441 Zk -flow, 437 2-objective optimization problem, 44 odd K4 , 429, 473 chain, 284 cycle, 12, 35 cycle through a fixed arc, 337 digraph (k, p)-odd digraph, 326 necklace, 473 orientation, 428, 430 one-way communication, 691 cut, 546 pairs, 214, 563–566, 581 set of arcs, 82 O, Ω, Θ-notation, 696 open pth in-neighbourhood, 88 open pth out-neighbourhood, 88 open problem, 121, 146, 206, 231, 268, 280, 292–295, 302, 303, 320, 325, 327, 329, 354, 356, 370, 379, 381, 399, 412, 429, 431, 435, 470–472, 491, 497, 500–502, 515, 518, 523, 524, 527, 547–549, 568, 575, 576, 597, 605, 613, 635, 647, 680, 694 opposite vertices, 283 1-OPT, 708, 720 optimal augmentation, 557 base of a matroid, 714 flow, 162 submodular flow, 459 optimization problem, 701 order Subject Index of a digraph, of functions, 696 order exchange, 612 order reflection, 612 ordered partitioning, 657 ordinary arc, 60, 295, 574 cycle, 60 path, 60 orientation, 473 as a local tournament, 422 as a quasi-transitive digraph, 418 as a transitive digraph, 418 as an in-tournament digraph, 428 best-balanced, 443 nowhere-zero flows, 435–441 odd, 428 of a graph, 417–453 respecting degree constraints, 448–453 smooth, 443, 475 strong, 20 of mixed graph, 201 well-balanced, 443 orientation of a digraph, 25, 85, 467, 468, 691 orientation of a graph, see also orientation, 20, 121, 201, 266, 268, 293, 349, 350, 365, 366, 417–453, 459, 460, 472, 473, 475–478, 631, 681, 682 minimum diameter, 103, 104, 691 with high arc-strong connectivity, 476 with small strong radius, 105 orientation of a mixed graph, 24, 452, 461–466 with small diameter, 107 orientation of a mixed multigraph, 225 orientation of a multigraph, 441 oriented cycle, 20 forest, 21 graph, 14 hamiltonian cycle in a tournament, 301–303 hamiltonian path, 297–301 path, 20, 298 tree, 21 CuuDuongThanCong.com 789 origin of an oriented path, 298 orthogonal rows in a matrix, 39 out-branching, see also in-branching, 22, 58, 339, 519, 705 arc-disjoint, 580 BFS tree, 93 minimum cost, 497 of shortest paths, 90 out-branchings min/maxleaf problems, 358–363 with bandwidth constraints, 367 k-out-critical set, 208 out-degree of a vertex, out-forest, 368 out-generator of a digraph, 299 out-neighbour, out-neighbourhood, pth out-neighbourhood, 88 out-path, 298 out-path-mergeable digraph, 58 out-pseudodegree of a vertex, out-radius, 89 finite in a weighted digraph, 89 out-singular vertex with respect to a cycle, 247 out-tree, 22, 346, 705 outer face of a plane (di)graph, 72 P, 700 P-gadget, 617–621 P-gadget graph, 618–621 packing cuts, 505 pancircular digraph, 316 pancyclic digraph, 307–316, 336 parallel architectures, 44 arcs, composition of digraphs, 48 reduction, 50 partial order, 521 comparable elements, 521 p-partite digraph, 34 p-partite graph, 19 partite sets, 19 partition, path, 12 (X, Y )-path, 12 (x, y)-path, 12 [x, y]-path, 12 xy-path, 21 anti-directed, 297 arc-disjoint, 201, 374 colourful, 322 crossing, 396 790 Subject Index edge-disjoint, 406 even, 12 finding a colourful path of prescribed length, 322 finding a path of prescribed length, 322 good reversal, 476 internally disjoint paths, 374 length, 12 longest, 12 odd, 12 of length Θ(log n), 321 ordinary, 60 oriented, 20 vertex-disjoint, 201, 374 path covering number, 15, 245, 258, 519, 521 path factor, 15, 277, 486, 520 path flow, 136 path partition conjecture, 542–546 q-path subdigraph, 15 path-contraction, 8, 310, 664 versus set-contraction, path-cycle covering number, 15, 245, 481, 519 path-cycle factor, 178, 303, 485 q-path-cycle factor, 15 q-path-cycle subdigraph, 14 path-mergeable digraph, 55–57, 85, 230, 231, 271, 287 hamiltonian (x, y)-path, 287 (s, t)-paths arc-disjoint, 201 internally disjoint, 201 PC, see properly coloured k-perfect family of hash functions, 323 perfect graph, 121 perfect matching, 21, 188, 451, 610, 615, 617–621, 626, 639, 640, 672 in a bipartite graph, 174 of minimum weight in a bipartite graph, 169 period of a directed pseudograph, 677 permutation graph, 83 PERT/CPM , 685–686 Petersen graph, 439, 473, 474 PFx problem, 277 planar digraph, 71–73, 224, 228, 384, 394–401, 404, 406, 407, 416, 511 feedback arc set problem, 590 CuuDuongThanCong.com linkage problem, 395 recognition, 72 vertex-strong connectivity of, 224 planar graph, 71 plane (di)graph, 72 point, 645 polygonal curve, 71 polymatroid, 368, 561 polynomial algorithm, 696 polynomial reduction, 701 polynomial-time approximation scheme (PTAS), 592, 593 power of a cycle, 224, 283, 303, 526 of a digraph, 10 of a hamilton cycle, 517 of a matrix, 337 of a path, 284, 303, 505 predecessor of a vertex on a path/cycle, 13 preflow, 150, 186, 347 preflow directed multigraph, 347 problem, 700 project scheduling, 685 projective plane, 645 proof technique BB-correspondence, 627 BD-correspondence, 641 colour-coding, 322 contraction, 506 DHM-construction, 630 divide and conquer, 699 gadgets for N P-completeness proofs, 228, 376, 551 insertion method, 697 iterative rounding, 500 matroid intersection, 461, 720 matroid partition, 720 multi-insertion, 239–243 one-way pairs, 214, 566, 567 probabilistic method, 328, 514, 548, 585, 637–640 random acyclic subdigraph method, 321 reduction to a flow problem, 202, 449, 579, 580 reversing arcs, 469 splitting off arcs, 559 splitting off edges, 441 submodular flows, 366, 459–466, 509 transversals in hypergraphs, 364 Subject Index uncrossing, 209, 507 using orientations of undirected graphs, 350 using recursive formulas, 45 using submodularity, 202, 346, 554, 555, 559 using the bipartite representation of a directed multigraph , 19 vertex splitting procedure, 201 proper circular arc graph, 418, 424 orientation as a round local tournament, 422 recognition in linear time, 423 colouring, 21, 431 edge-colouring, 680 interval graph, 115, 472 subset, properly coloured m-path-cycle subgraph, 608 1-path-cycle subgraph, 620, 621 cycle, 613, 620 cycle subgraph, 608, 618–620 Euler trail, 610 hamiltonian cycle, 621, 635–640 hamiltonian path, 635 path, 620, 621 trail, 608 pseudograph, 18 pseudoregular directed pseudograph, 44, 47 push-relabel algorithm, 150–152, 181 quasi-kernel, 119, 122 quasi-transitive digraph, 52–55, 108– 111, 118, 257, 258, 265, 272, 273, 309, 311, 336, 357, 392, 394, 416, 418, 420, 421, 469, 485, 486, 503 hamiltonian cycle, 256–259 highly connected orientation of, 469 minimum cycle factor, 521–524 MSSS problem, 485 path-partition problem, 544– 546 recursive characterization, 54 vertex-heaviest paths and cycles, 260–265 quasi-transitive orientation, 418 queue, 92 radius, 89, 100–101 CuuDuongThanCong.com 791 Ramsey’s theorem, 598 random acyclic subdigraph method, 321 rank function of a matroid, 712 rank of a matroid, 712 re-weighting the arcs of a digraph, 124, 125, 344 reachable from a vertex, 15 recognition interval digraphs, 83 line digraph, 41 path-mergeable digraph, 56 planar digraph, 72 round decomposable locally semicomplete digraph, 65 round local tournament digraphs, 472 totally Φ-decomposable digraph, 70 vertex series-parallel digraph, 52 red/blue subgraph of a 2-edgecoloured multigraph, 608 R´edei’s theorem, 14, 298 reducible graph, 424 reduction among flow models, 131 redundant arc of a digraph, 37 reference orientation, 448, 459 regular digraph, 5, 46, 102, 225, 228, 254, 525, 536 arc-disjoint cycles in, 513 regular graph, 20 regular oriented graph, 103 reorienting arcs, 459 representation of a digraph, 82 of a graph, 418 residual network, 130, 137 reversal of a path, 476 reverse of a trail, 609 reversing an arc, reversing arcs, see arc reversal reversing arcs to obtain arc-disjoint branchings, 478 Road Colouring Conjecture, 688 Robbins’ theorem, 20, 201, 441, 452 root of a branching, 22 of a DFS tree, 27 round decomposition, 62 digraph, 60–61, 332 labelling, 60 792 Subject Index round decomposable locally semicomplete digraph, 62–65 routing problems, 414 SAT, 703 2-SAT, 666–670, 702–703 application to orientability as in-tournaments, 426 3-SAT, 667, 703 Satisfiability, see also SAT, 667 satisfiable boolean expression, 667, 703 saturated arc, 144 scaling algorithm for maximum flow, 183 scan register, 583 scheduling problems, 47, 186, 520 score of a vertex, 449 score sequence, 449 semi-degree of a vertex, semi-partitioncomplete digraph, 256 semicomplete p-partite digraph, see semicomplete multipartite digraph semicomplete bipartite digraph, 35, 86, 108–111, 246, 252, 272, 276, 277, 337, 525, 528, 627, 628 even pancyclic, 336 hamiltonian cycle, 246 hamiltonian path with one end vertex specified, 276 longest cycle, 246 semicomplete decomposition of a locally semicomplete digraph, 63, 67 semicomplete digraph, 35, 53, 54, 115, 121, 126, 188, 225, 271, 283, 286, 287, 290, 291, 295, 296, 304, 314, 320, 385, 386, 388–391, 393, 394, 407, 410, 414, 468, 469, 516, 528, 548, 551, 567–569 2-linkage problem, 389 hamiltonian (x, y)-path, 286 hamiltonian connected, 287 hamiltonian-connected, 287 highly connected orientation of, 468 vertex-heaviest paths and cycles, 260–265 semicomplete multipartite digraph, 35, 71, 111, 116–118, 244, CuuDuongThanCong.com 245, 247, 249, 251, 253–256, 272, 273, 295, 315, 316, 481, 525, 526 ‘short’ cycles, 332–335 hamiltonian cycle, 244–256 hamiltonian path, 245 longest path, 245 path covering number, 245 regular, 254 separator, 16 (s, t)-separator, 17 minimum, 192 trivial, 287, 391 sequencing problems, 47 2-serf, 116 series composition of digraphs, 48 series reduction, 50 series-parallel digraph, 47–52 k-set, set covering problem, 582 set-contraction, see contraction Seymour’s second neighbourhood conjecture, 600 ship loading problem, 161 short cycle in a digraph, 321 shortest cycle, see also girth, 125 path tree form s, 90 paths, 90, 130 k-similar arms of chromosomes, 671 similar size arms of chromosomes, 670 similar vertices, 10, 407 simple network, 155 simulated annealing, 709–711 singular vertex, 247 sink of a network, 129 of an anti-directed trail, 215 vertex with respect to a flow, 129 sink in a digraph, size of a digraph, smooth orientation, 443, 475 solution of an optimization problem, 708 sorting versus distances in digraphs, 97 source in a digraph, of a network, 129 of an anti-directed trail, 215 vertex with respect to a flow, 129 Subject Index source location problem, 363 source-sink connected digraph, 511 spanning out-forest, 368 spanning tree, 21 specific trail (ST) problem, 402, 405 Sperner’s lemma, 126, 547 splitting, 554 vertices, see vertex splitting procedure splitting a vertex, 10 splitting off, 475, 553–562, 580 admissible, 554 complete, 556 in eulerian directed multigraphs, 557 in mixed graphs, 557 in undirected graphs, 441 in undirected multigraphs, 442 preserving local arc-strong connectivity, 557 splitting off arcs, 213 stable matching, 681 star hypergraph, 577, 607 state diagram, 687 Steiner tree problem, 366 straight digraph, 472 ordering, 472 strictly alternating cycle, 636 strong k-strong, 16 digraph, 16, 195–198 orientation, 20, 201, 452 k-strong augmentation number of a digraph, 563, 565 strong component digraph, 17 strong components, 17, 232, 668 algorithm for finding, 195 application to finding blocktriangular structure in matrices, 676 strong decomposition of a digraph, 17, 59 Strong Perfect Graph theorem, 122 strong radius, 105 strongly connected, see strong strongly polynomial algorithm, 165 subdigraph, spanning, see also factor of a digraph, with prescribed degrees, 176 minimum cost, 176 subdivision CuuDuongThanCong.com 793 of a digraph, 10, 411 of an arc, 10, 326, 483, 507 submodular flows, 453–466, 477, 478, 509, 570, 582 submodular function, 193 minimizing, 457 submodularity of (s, t)-cuts, 185 of matroid rank functions, 713 subpartition, subpartition lower bound, 558, 563, 565 subpath, 13 subtree intersection digraph, 83 successive arc-connectivity augmentation property, 561 successor of a vertex on a path/cycle, 13 sum of boolean variables, 667, 702 superdigraph, supermodular function, 448, 455 G-supermodular function, 452 switch, 376 (e, f )-switch, 672 symmetric design, 645 digraph, 20 function, 452 synchronizing string, 688 system of distinct representatives (SDR), 645 Szemer´edi regularity lemma directed version, 512 tail of a one-way pair, 563 of an arc, telecommunications, 414, 499 terminal strong component, 17 terminal vertex of a walk, 12 terminals of a trail in eulerian directed multigraph, 402 terminus of an oriented path, 298 Thomassen’s even cycle theorem, 326 tight arc, 456 set, 202, 456 Tillson’s decomposition theorem, 516 time complexity of an algorithm, 696 topological obstruction for linkages, 394 topological sorting, see acyclic ordering total unimodularity, 535 794 Subject Index totally Φ-decomposable digraph, 10, 52, 69–71 hamiltonian cycle, 259 hamiltonian path, 259 recognition, 70 total Φ-decomposition, 10 totally unimodular matrix, 145 tournament, see also semicomplete digraph, 14, 84, 104, 115, 117, 122, 126, 188, 218, 224, 225, 278, 279, 286–288, 290, 292, 293, 297–299, 301, 302, 304, 305, 317, 318, 336, 356, 357, 385, 390–392, 407, 415, 449, 467–469, 473, 492, 516–518, 527, 548, 549, 552, 567, 569, 580, 581, 592, 601, 602, 697, 709 arc-3-cyclic, 318, 517 complementary cycles, 525, 549 decomposition into strong subdigraphs, 536 feedback vertex set problem, 587 hamiltonian [x, y]-path, 277 weakly hamiltonian-connected, 277 traceable, see also hamiltonian path, 13, 14, 236, 530, 697 arc-traceable digraph, 296 trail, 12 alternating, 608 M -trail, 610 transitive closure, 37, 38 versus transitive reduction, 38 digraph, 36, 54, 85, 521, 533 reduction, 37, 48 tournament, 84, 597, 719 triple, 30 transputer-based machine, 691 transversal of a hypergraph, 364 travelling salesman problem, see TSP tree, 21 tree decomposition, 73–77 tree solution to a flow problem, 188 tree-width, 73–77 triangular digraph, 309, 311 trivial (s, t)-separator, 287 trivial separator, 391 truth assignment, 667, 703 TSP, 655, 658–666, 694, 700, 705, 707 k-tuple, 382 Tutte’s 5-flow conjecture, 439, 475 CuuDuongThanCong.com two-terminal parallel composition, 48 series composition, 48 unbalanced edge, 424 uncrossing technique, 209, 507 underlying graph, 418–428, 467 underlying graph of a digraph, see also underlying graph, 20, 27, 53–55, 70, 85, 201, 231, 294, 343, 349, 384, 396, 407, 417–428, 431, 432, 446, 467, 472, 509, 536, 681, 717 underlying multigraph of a digraph, 20 undirected graph, 18 non-critical edge of, 217 uniform independence system, 657 matroid, 713 unilateral digraph, 17 union of digraphs, 11, 38 of matroids, 715 unique trail (UT) problem, 402, 404 unit capacity network, 153 universal arc, 412 set, 82 k-universal digraph, 432 upward embedding, 85 value of a flow, 132 of a solution, 708 vertex, cost, weight, vertex cover of a bipartite graph, 172 vertex cover problem (VC), 172, 704 vertex even pancyclic digraph, 336 vertex separation, 78 vertex series-parallel (VSP) digraph, 48 vertex series-parallel digraph recognition algorithm, 52 vertex splitting, 202 vertex splitting procedure, 134, 202, 398, 512, 685 vertex-arc incidence matrix, 145 vertex-cheapest k-path subdigraph, 261 cycle, 261 Subject Index vertex-pancyclic digraph, 307, 309, 311, 313, 314 vertex-strong connectivity, 17, 191– 226 algorithms, 204 certificate, 490 of complete biorientations, 225 of extensions of digraphs, 220 of special classes of digraphs, 220 vertex-weighted directed pseudograph, Volkmann’s meta-conjecture, 526 W[1] problems, 704 W[1]-complete, 704 W[1]-hard, 704 walk, 11–13 (x, y)-walk, 11 Chinese postman walk, 174 weak k-linkage problem, 398–414 semicomplete digraphs, 409 weak linkages, 373, 374, 398–414 acyclic digraphs, 400 eulerian directed multigraphs, 401–407 CuuDuongThanCong.com 795 generalizations of tournaments, 407–414 k-weak-double-cycle, 326, 337 weakly k-linked digraph, 398, 399 directed multigraph, 373 weakly cycle extendable, 336 weakly hamiltonian-connected, 277– 285, 289 weight of a subdigraph, 6, 260 of a vertex, of an arc, weighted arc-strong connectivity augmentation problem, 561 weighted directed pseudograph, well-balanced orientation, 443 Woodall’s conjecture, 511 Yeo’s irreducible cycle subdigraph theorem, 253, 293 Younger’s conjecture, 597–600 Zemel measure, 707 ... often use the shorthand notation (H, L)D , H→L, H⇒L and H→L instead of (V (H), V (L))D , V (H)→V (L), V (H)⇒V (L) and V (H)→V (L) A weighted directed pseudograph is a directed pseudograph D along... coincide (i.e., V (D) = V (H) and A(D) = A(H )) In particular, there are four labeled digraphs with vertex set {1, 2} Indeed, the labeled digraphs ({1, 2}, {(1, 2)} ) and ({1, 2}, {(2, 1)} ) are distinct,... (maximum) degree of G is δ(G) = min{d(x) : x ∈ V (G)} (Δ(G) = max{d(x) : x ∈ V (G) }) We say that G is regular (or δ(G)-regular) if δ(G) = Δ(G) A pair of ↔ ↔ graphs G and H is isomorphic if G and