Graph Relations and Constrained Homomorphism Partial Orders Von der Fakultät für Mathematik und Informatik der Universität Leipzig angenommene DISSERTATION zur Erlangung des akademischen Grades DOCTOR RERUM NATURALIUM (Dr.rer.nat.) im Fachgebiet Mathematik vorgelegt von Bachelorin Yangjing Long geboren am 08.07.1985 in Hubei (China) Die Annahme der Dissertation wurde empfohlen von: 1. Professor Dr. Pavol Hell (Burnaby, Kanada) 2. Professor Dr. Jürgen Jost (Leipzig) Die Verleihung des akademischen Grades erfolgt mit Bestehen der Verteidigung am 14.10.2014 mit dem Gesamtprädikat magna cum laude. 献给我挚爱的母亲们！ Acknowledgements My sincere and earnest thanks to my ‘Doktorväter’ Prof. Dr. Peter F. Stadler and Prof. Dr. Jürgen Jost. They introduced me to an interesting project, and have been continuously giving me valuable guidance and support. Thanks to my co-authers Prof. Dr. Jiří Fiala and Prof. Dr. Ling Yang, with whom I have learnt a lot from discussions. I also express my true thanks to Prof. Dr. Jaroslav Nešetřil for the enlightening discussions that led me along the way to research. I would like to express my gratitude to my teacher Prof. Dr. Yaokun Wu, many of my senior fellow apprentice, especially Dr. Frank Bauer and Dr. Marc Hellmuth, who have given me a lot of helpful ideas and advice. Special thanks to Dr. Jan Hubička and Dr. Xianqing Li-jost, not only for their guidance in mathematics and research, but also for their endless care and love—they have made a better and happier me. Many thanks to Dr. Danijela Horak helping with the artwork in this thesis, and also to Dr. Andrew Goodall, Dr. Johannes Rauh, Dr. Chao Xiao, Dr. Steve Chaplick and Mark Jacobs for devoting their time and energy to reading drafts of the thesis and making language corrections. I am very grateful to IMPRS for providing a financial scholarship for my studies and to Research Academic Leipzig for supporting my attendance at conferences. I also wish to express my appreciation to the administrative staff at MIS MPG and Mrs. Petra Pregel for their supportiveness. Finally, to my family and friends for their ongoing care, support and encouragement—you make my life colourful. Abstract We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective ho- momorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every d ≥ 3 the homomorphism order on the class of line graphs of graphs with maximum degree d is universal. Contents Contents v List of Figures ix 1 Introduction 1 2 Graphs and Orders 9 2.1 Graphs.................................. 9 2.1.1 Basicconcepts.......................... 9 2.1.2 Graphconstructions . 13 2.1.3 Graphclasses .......................... 16 2.1.4 Matrixrepresentationsofgraphs . 21 2.2 Orderedsets............................... 24 2.2.1 Basicdefinitions. 25 2.2.2 Functions between orders . 26 2.2.3 Basicproperties. 27 3 Homomorphisms and Constrained Homomorphisms 31 3.1 Graphhomomorphisms. 31 3.1.1 Homomorphisms preserve adjacency . 34 3.1.2 Homomorphism equivalence . 34 3.1.3 H-colouringproblem . 36 3.2 Globally constrained homomorphisms . 37 3.2.1 Surjective homomorphisms . 38 3.2.2 Fullhomomorphisms . 39 3.3 Locally constrained homomorphisms . 42 3.4 Relations................................. 47 vi CONTENTS 3.4.1 Matrixmultiplication. 49 3.4.2 Homomorphisms and multihomomorphisms . 49 3.4.3 Examples ............................ 50 4 R-Homomorphisms (Relations) 53 4.1 Basicproperties............................. 54 4.1.1 Composition........................... 54 4.1.2 Structural properties preserved by relations . 55 4.2 Relationalequivalence . 60 4.2.1 Reversiblerelations . 60 4.2.2 Strong relational equivalence . 61 4.2.3 Weak relational equivalence . 62 4.2.4 R-cores.............................. 63 4.3 The partial order Rel(G,H) ...................... 68 4.3.1 Basicproperties. 68 4.3.2 Solutions of G ∗ R = G ..................... 68 4.4 R-Retractions .............................. 71 4.4.1 Cocores ............................. 73 4.4.2 Inclusionrelation . 78 4.5 Computationalcomplexity . 79 4.6 Weakrelationalcomposition . 80 5 Constrained Homomorphisms Orders 83 5.1 Universalorders............................. 84 5.2 Properties of past-finite and future-finite orders . ...... 88 5.3 Homomorphismorders . 91 5.3.1 Universality ........................... 92 5.3.2 Density and duality . 92 5.3.3 Restricted homomorphism universality . 94 5.4 Embedding and Monomorphism orders . 96 5.5 Fullhomomorphismorders . 97 5.6 Surjective homomorphism orders . 102 5.7 Locally constrained homomorphism orders . 104 5.7.1 Orders of degree refinement matrices . 105 5.7.2 Locally surjective and locally bijective orders . ..... 106 5.7.3 Locally injective homomorphism orders . 108 5.7.4 RemarksontheordersofDRMs. 114 5.8 Theintervalsofthelinegraphorder . 114 5.8.1 Dragongraphs ......................... 116 5.8.2 Indicatorconstruction . 119 5.8.3 Finalconstruction . 120 CONTENTS vii 5.8.4 Concludingremarks. 122 5.9 TheordersofRelations. 124 5.9.1 PR-cores............................. 125 5.9.2 Properties of (DiGraphs, ≤P R) and (DiGraphs, ≤R) ...... 125 5.10 ConcludingRemarks . 127 Appendix A: Computational complexity 129 References 133 Index 140 List of Figures 2.1 Directedgraphs.............................. 10 2.2 A directed graph and its underlying graph. 11 2.3 Differentclassesofgraphs. 11 2.4 Disconnected graph with an isolated vertex and a loop. ..... 12 2.5 Subgraph and induced subgraph. 12 2.6 An example of walk, path and cycle. 13 2.7 Petersengraph ............................. 14 2.8 Vertex deletion Gu = P4......................... 15 2.9 Vertex multiplication: P4 ◦ (2, 1, 2, 1).................. 15 2.10 Vertex contraction {1, 3} and {1′, 2}. ................. 16 2.11 Quotient graph of Frucht graph (left) is K3. ............. 16 2.12 Directedpath,orientedpathandpath. 17 2.13Cyclegraphs. .............................. 18 2.14 Oriented cycle with 8 vertices. 18 2.15 Adirectedacyclicgraph. 18 2.16 Completegraphs. .. .. .. 19 2.17 Acomplete3-partitegraph. 19 2.18Trees. .................................. 20 2.19Anexample................................ 22 2.20Matching. ................................ 23 2.21 Perfectmatching. 23 2.22Hassedigrams. ............................. 26 2.23 Directedacyclicgraph. 27 3.1 Aseriesofretractions. 35 3.2 Monomorphism, surjective homomorphism, embedding. ..... 37 x LIST OF FIGURES 3.3 Example of an edge surjective but not vertex surjective homomor- phism. .................................. 38 3.4 A full homomorphism from G to H................... 40 3.5 Locally injective, bijective, surjective homomorphisms [22]. ..... 43 3.6 H and its universal covering graph. 47 4.1 Commutativediagram. 55 4.2 There is no relation from G to H.................... 59 4.3 Non-isomorphic graphs G and H with isomorphic point-determining graphs................................... 62 4.4 G and H are weakly relationally equivalent but have non-isomorphic graphs................................... 63 4.5 Commutativediagram. 64 4.6 Construction of an embedding from GR-core to G. .......... 65 4.7 A graph G anditscore. ........................ 73 4.8 Agraphanditscocore.. 74 4.9 Inclusiondigram ............................ 79 5.1 The order (P, ≤P ). Blue lines represent backwarding edges . 88 5.2 Generalized duality pair ({f1, f2}, {d1,d2})............... 89 5.3 Dualitiesinafuture-finiteorder.. 90 5.4 Dualitypair................................ 93 5.5 Graph G with vertex u extended by a (5,11)-lasso. 109 5.6 The order (P, ≤P ). ........................... 112 5.7 Representation of order (P, ≤P ). Black circles denote left sunlets, blue circles right sunlets. 112 5.8 The 3-dragon D3 and its line graph L(D3)............... 117 5.9 Line graph of 4-dragon with cliques corresponding to the neighbour- hoods of two vertices distinguished. 118 5.10 Mapping two d-cliquestothesametarget. 118 5.11 Indicator I3(a, b) anditslinegraph. 119 5.12 The 5-sunlet S5.............................. 120 5.13 The graph G5,3 = S5 ∗ I3(a, b) anditslinegraph.. 121 Chapter 1 Introduction The subject of this thesis—graph relations and constrained homomorphism orders—belongs to the field of discrete mathematics, more specifically to graph theory. It has a close relationship to several other areas of mathematics stemming from algebra and mathematical structures via category theory and logic to computer science disciplines such as computational complexity. The term ‘graph’ was introduced by Sylvester in 1878 [79]. A directed