Structural Aspects of Switching Classes Structural Aspects of Switching Classes Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magni¯cus Dr. D.D. Breimer, hoogleraar in de faculteit der Wiskunde en Natuurwetenschappen en die der Geneeskunde, volgens besluit van het College voor Promoties te verdedigen op woensdag 26 september 2001 te klokke 14.15 uur Jurriaan Hage geboren te Alphen aan den Rijn in 1969 Promotiecommissie Promotor: Prof. dr. G. Rozenberg Co-promotor: Dr. T. Harju (Turun Yliopisto, Finland) Referent: Prof. dr. H.-J. Kreowski (UniversitÄat Bremen, Duitsland) Overige leden: Prof. dr. H. Ehrig (Technische UniversitÄat Berlin, Duitsland) Prof. dr. E. Welzl (ETH ZÄurich, Zwitserland) Dr. J. Engelfriet Prof. dr. J. Kok Prof. dr. H.A.G. Wijsho® The work in this thesis was carried within the context of the IPA graduate school. And then there was light... Contents 1 Introduction 11 2 Preliminaries 15 2.1 Sets, functions and relations ....................... 15 2.2Basicgrouptheory............................ 16 2.3Basicgraphtheory............................ 22 I Switching Classes of Graphs 27 3 Switching Classes 29 3.1De¯nitions................................. 29 3.2Basicpropertiesofswitchingclasses.................. 32 3.3 Two-graphs . ............................... 37 3.4Somecomplexityconsiderations..................... 37 3.4.1 Easyproblemsforswitchingclasses............... 38 3.4.2 Hardproblemsforswitchingclasses.............. 40 3.4.3 Theembeddingproblem..................... 43 4 Cyclicity Considerations 47 4.1Pancyclicityinswitchingclasses..................... 47 4.2Countingacyclicgraphsinswitchingclasses.............. 49 4.2.1 Preparation............................ 50 4.2.2 Trees . ............................... 51 4.2.3 Trees into disconnected acyclic graphs . ........... 52 4.2.4 Disconnectedacyclicgraphs................... 54 4.3Characterizingacyclicswitchingclasses................ 57 4.3.1 Thespecialgraphs........................ 59 4.3.2 Isolatedvertices......................... 64 4.3.3 Thecases............................. 68 4.3.4 Thecase(1s)........................... 69 4.3.5 Theothercases.......................... 70 4.3.6 Thecases(2s)-(4s)........................ 71 4.3.7 Thecases(5s)-(8s)........................ 71 4.3.8 Concludingremarks....................... 72 7 8 II Switching Classes of Graphs with Skew Gains 75 5 Gain Graphs 77 5.1De¯nitions................................. 77 5.2Anti-involutions.............................. 82 5.3Anti-involutionsofcyclicgroups.................... 85 5.4Spanningacyclicskewgainsubgraphs................. 87 6 The Sizes of Switching Classes 91 6.1Completegraphswithskewgains.................... 91 6.2Improvementsinsomespecialcases................... 96 6.3Thegeneralcase............................. 98 7 The Membership Problem 103 7.1Thegeneralproblemofmembership.................. 103 7.2Algorithms................................ 106 7.3Improvementsintheabeliancase.................... 108 7.3.1 Improvements when gT hasabeliangains........... 108 7.3.2 Thesetofskewedsquares.................... 109 7.3.3 Unre¯nableswitchingclasses.................. 111 7.4 Undecidability for arbitrary groups ................... 113 8 Future Directions 115 8.1Problemsforswitchingclassesofgraphs................ 115 8.2Problemsforswitchingclassesofskewgaingraphs.......... 116 A Algorithms and Programming Techniques 119 A.1Enumeratingsubmultisets........................ 119 A.2 An example: the case m =2....................... 123 A.3 The computation of the switches of a graph . ........... 127 A.4Switchingskewgaingraphs....................... 129 B Researching Switching Classes With Programs 135 B.1 The Scheme programs.......................... 136 B.2 The C programs.............................. 137 B.3 A Java-applet............................... 138 Preface This thesis studies the subject of switching classes of undirected graphs and switching classes of skew gain graphs. The ¯eld of research lies in graph theory with a group theoretical avour. A switching class is an equivalence class of graphs under the switching operation, that leaves the set of vertices unharmed, but may destroy or create new edges. In the ¯eld of switching classes of skew gain graphs the edges have a label from a group. The subjects that are studied here are often of a combinatorial nature and in- vestigate diverse properties of switching classes. There is a rather strong focus on algorithms, which are based on the theory presented here. The thesis also contains a number of complexity results. The initial motivation for the subject came, in our case, from a model of net- works of processors of a speci¯c kind: processors that by an action inuence all the connections they might have to other processors. Results in this ¯eld are part of the second part of the thesis. The results focus on the study of the evolutionary behaviours of such networks of processors and algorithms to make certain problems decidable and, once decidable, feasible. During the research we found that the model we used was already de¯ned in a simpler form in mathematics. It turned out however that many of our results were new which shows the diversity in motivation between those ¯elds. The ¯rst part of the thesis is of a more graph theoretic nature and concerns itself with the occurence of special kinds of graphs such as acyclic graphs, hamiltonian graphs and the like in switching classes. Another motivation for our research has to do with problems that are di±cult for graphs, e.g., NP-complete problems. In our research we found that the hamiltonian cycle problem is rather simple for switching classes, while we know it is hard for graphs. Instead of answering a di±cult question for a speci¯c graph, we might ¯rst approximate the answer by answering the question for the switching class in which that graph resides. Of a more theoretical nature are the results that link graph theory with the theory of switching classes. For instance, it is proved in this thesis that switching classes contain at most one tree up to isomorphism. In such a way we also learn more about the nature of these types of graphs. Much of the material in this thesis comes from articles that appeared in various places. Examples have been added to clarify the material from the papers. In some cases generalizations replace the results of the paper and new results have been added as well; in all cases we indicate where such is the case. For completeness we give a general overview of what ended up where: the results about complexity by Ehrenfeucht, Hage, Harju and Rozenberg [12] can be found 9 10 in Section 3.4. A related result about pancyclic graphs in switching classes is by Ehrenfeucht, Hage, Harju and Rozenberg [13] and appears in Section 4.1. The two papers [24] and [25] by Hage and Harju about trees and acyclic graphs in switching classes can be found in the same chapter, Sections 4.2 and 4.3. For the skew gain graphs, Chapter 6 is based on [26] by Hage and Harju and Chapter 7 contains the material of Hage [21]. The results of Appendix A can be of interest to people programming switching classes. Many people at LIACS helped me during my work, or indulged in my desire to propound my newest results and latest optimizations. In between writing papers and program, I have greatly enjoyed the long discus- sions with Frans Birrer, Joost Engelfriet, Jano van Hemert, Hendrik Jan Hooge- boom, but also the shorter ones with Walter Kosters, Marloes van der Nat, Rudy van Vliet, Pier Frisco and Henk Goeman. I owe a great debt to Maurice ter Beek, Sebastian Maneth and most of all to Tjalling Gelsema for carrying the burden of sharing a room with me. Financially the work was supported (in part) by Arto Salomaa, Ralph Back (and all at TUCS) and the GETGRATS project. I am grateful to LIACS for the extra time granted me in ¯nishing the thesis and also gaining some experience in lecturing. I am grateful to Piet van Oostrum and Tycho Strijk for help with LATEXand CorelDraw. On the personal front I thank my parents and my brother Arjan for their en- couragement. I am grateful to Roberto Lambooy for his driving support. This is also a good place to mention Jaap van der Sar, Joop Minderman, Frits Feddes, Jan van de Sant, Piet van Tienhoven and Leo van de Heuvel from the energy company (G.E.B./E.W.R./Nuon) in Alphen aan den Rijn for, as they still constantly remind me, "teaching me everything I know". Finally, I want to thank Nike, for her constant encouragement, inspiration and doing the household chores during the busy periods. I hope to return the favour in the near future. Jurriaan Hage Alphen aan den Rijn, May 2001. Chapter 1 Introduction This thesis covers a number of problems in the area of switching classes of undirected graphs and that of switching classes of directed graphs with skew gains. It is a self contained excursion into these areas. An understanding of group theory, graph theory and the theory of algorithmic complexity is presumed, but in the ¯rst two cases, we do establish terminology and notation. Knowledge of group theory is only necessary for the second part of the thesis which treats switching classes of graphs with skew gains. We introduce the switching classes of graphs of the ¯rst part of the thesis: for a ¯nite undirected graph G =(V;E) and a subset V (called a selector), the vertex-switching of G by is de¯ned as the graph G =(V;E0), which is obtained from G by removing all edges between and its complement