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Reconstructing a Graph from Its Edge-Contractions

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Reconstructing a Graph from Its Edge-Contractions Reconstructing a graph from its edge-contractions Antoine Poirier Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics1 Department of Mathematics and Statistics Faculty of Science University of Ottawa c Antoine Poirier, Ottawa, Canada, 2018 1The Ph.D. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract In this thesis, we investigate the contraction reconstruction conjecture. It states that all simple graphs with at least four edges are reconstructible, that is they are uniquely determined from their collection of single edge contraction minors, called the deck. Similar questions have been studied in the past, the vertex reconstruction conjecture being the most famous. There are usually two steps to show that a class of graph is reconstructible. The first one is to show that the class is recognizable, meaning that it is possible to determine if a graph G belongs to that class by looking at its deck. In order to recognize some classes of graphs, we show that a wide range of graph properties are reconstructible. We investigate the connectivity of graphs, which is useful to recognize disconnected, separable, and 2-connected graphs. We also show that the number of cycles of various lengths, the degree sequence, the number of spanning trees, the planarity, the presence of cliques of various sizes, and the diameter are reconstructible. Knowing the lengths of cycles allows us to recognize the class of bipartite graphs, while knowing the degree sequence allows us to recognize regular graphs. The second step in showing that a class of graph is reconstructible is called weak reconstruction. We say that a class of graph is weakly reconstructible if no two graphs in that class share the same deck. A class of graphs that is both weakly reconstructible and recognizable is reconstructible. In this thesis, we show that disconnected graphs, bipartite graphs, most separable graphs and most 2-edge connected graphs are re- constructible. We also show that distance regular graphs and some cubic graphs are reconstructible. We quickly delve into the theory of probabilities to give a proof that almost all graphs are reconstructible. Finally, the relation between edge contraction and graph automorphisms is stud- ied. We study the automorphism group of a graph in relation to those of its cards. We also study the concept of contraction pseudo-similarity. Two edges are contraction pseudo-similar if they are not similar, but their contractions yield isomorphic graphs. We completely characterize the graphs that contain contraction pseudo-similar edges. ii R´esum´e Dans cette th`ese,la conjecture de la reconstruction de graphes par contraction est ´etudi´ee. Elle ´enonceque tous les graphes simples avec au moins quatre ar^etessont reconstructibles, c’est-`a-direqu'ils sont d´etermin´espar l'ensemble de leurs mineures obtenus par une contraction. Cet ensemble se nomme le deck. Des questions similaires ont ´et´e´etudi´eesdans le pass´e,la conjecture de la reconstruction de graphes par effacement de sommets ´etant la plus c´el`ebre. La reconstruction d'une famille de graphes comporte deux ´etapes. La premi`ere est la reconnaissance. Il faut montrer qu'il est possible de d´eterminersi un graphe appartient `acette famille en ne consid´erant que son deck. Dans cette th`ese,nous montrons que plusieurs propri´et´esd'un graphe sont reconstructibles. Nous ´etudions la connectivit´e,ce qui est utile pour reconna^ıtredes graphes d´econnexes,s´eparables et 2-connexes. Nous montrons aussi qu'il est possible de reconstruire le nombre de cycles de diff´erentes tailles, la s´equencedes degr´es,le nombre d'arbres couvrants, la planarit´e,le nombre de cliques de diff´erentes tailles ainsi que le diam`etre. La reconstruction de certains cycles nous permet alors de reconna^ıtreles graphes bipartis, alors que la reconstruction de la s´equencedes degr´esnous permet de reconna^ıtreles graphes r´eguliers. La deuxi`eme´etape dans la reconstruction est la reconstruction faible. Une classe de graphes est faiblement reconstructible si aucune paire de graphes dans cette classe ne partagent le m^emedeck. Une classe de graphes qui est `ala fois reconnaissable et faiblement reconstructible est reconstructible. Dans cette th`ese,nous montrons que les graphes d´econnexes, bipartis, s´eparableset plusieurs graphes 2-connexes sur les ar^etessont reconstructibles. Nous montrons aussi que les graphes distance-r´eguliers et quelques graphes cubiques sont reconstructibles. Nous empruntons aussi quelques ´el´ements de la th´eoriedes probabilit´espour montrer que presque tous les graphes sont reconstructibles. Finalement, la relation entre la reconstruction par contraction et les groupes d'automorphismes de graphes est ´etudi´ee.Nous ´etudionsle groupe d'automorphisme d'un graphe et sa relation avec ceux de ses cartes. Nous ´etudionsaussi la pseudo- similarit´e.Deux ar^etessont pseudo-similaires si elles ne sont pas similaires, mais que leurs contractions donnent deux graphes isomorphes. Nous caract´erisonsles graphes qui poss`edent de telles ar^etes. iii Remerciements Je voudrais d'abord remercier mon superviseur, Mike Newman. Sans sa pers´ev´erance, son d´evouement et ses bonnes id´ees,je n'aurais pas eu autant de succ`esni de plaisir `acompl´eterce doctorat. J'aurai toujours des bons souvenirs de nos rencontres heb- domadaires. Je tiens aussi `aremercier Gena Hahn, Lucia Moura, Monica Nevins, Daniel Panario, Mateja Sajnaˇ et Brett Stevens pour leurs commentaires et suggestions. Les conf´erences`al'ext´erieurd'Ottawa n'auraient pas ´et´eaussi plaisantes sans la pr´esenced'Elizabeth Maltais. Je remercie les membres du personnel du d´epartement de math´ematiquespour tous les soucis desquels ils m'ont lib´er´e,et Joseph Khoury pour s'^etreoccup´edu centre d'aide en math´ematiques,auquel j'ai travaill´eles six derni`eresann´ees. Finalement, je dis merci `ama famille et mes amis pour leur appui constant et leurs encouragements. Je suis tr`esreconnaissant envers ma copine, Claudine Bouvier. Je n'aurais jamais franchi toutes les ´epreuves qui se sont pr´esent´eesdans les derni`eres ann´eessans elle. Ce doctorat est autant le sien que le mien. iv Contents List of Figures ix 1 Introduction 1 1.1 Reconstruction . .1 1.2 Graph definitions . .4 2 A survey of reconstruction problems 7 2.1 Vertex reconstruction . .7 2.2 Edge reconstruction . 10 2.2.1 Definitions and elementary results . 10 2.2.2 Almost all graphs are edge reconstructible . 12 2.2.3 Pseudo-similarity . 13 2.3 Switching reconstruction . 15 3 Preliminary results on contraction reconstruction 17 3.1 Elementary results . 17 3.2 Contraction version of Kelly's Lemma . 21 3.3 Cliques and independent sets . 23 4 Connectivity 26 4.1 Vertex connectivity . 26 4.1.1 Definitions and elementary results . 26 4.1.2 Separable graphs . 28 4.1.3 2-connected graphs . 31 4.2 Reconstructing disconnected graphs . 33 4.2.1 Definition and main result . 33 4.2.2 Reduction reconstruction . 34 4.3 Reconstructing separable graphs . 36 4.3.1 Trees . 36 4.3.2 Blocks of separable graphs . 40 4.3.3 Separable graphs with at least two non-trivial blocks . 44 4.3.4 Separable graphs with one non-trivial block . 46 v CONTENTS vi 4.4 Edge connectivity . 53 4.4.1 Definitions and elementary results . 53 4.4.2 Edge blocks . 56 4.5 Reconstructing graphs with κ0(G)=2............... 62 4.6 Tutte polynomial . 68 5 Cycles 74 5.1 On the number of cycles . 74 5.1.1 Loops and parallel edges . 74 5.1.2 Girth . 77 5.1.3 Reconstructing the number of cycles . 77 5.1.4 Closed trails . 79 5.2 Bipartite graphs . 80 5.3 Planar graphs . 87 5.3.1 Maximal planar graphs . 93 6 Degrees 96 6.1 Reconstruction of the degree sequence . 96 6.2 Regular graphs . 100 6.2.1 1-regular and 2-regular graphs . 100 6.2.2 Cubic graphs . 101 6.3 Distance regular graphs . 103 6.3.1 Strongly regular graphs . 103 6.3.2 Distance regular graphs . 109 7 Automorphisms 117 7.1 Edge-automorphisms . 117 7.2 Pseudo-similarity . 121 7.2.1 Pseudo-similar edges in full paths . 123 k 7.2.2 If θ(vb) 6= va and θ (vb) = va for some k ≥ 2. 124 7.2.3 If θ(vb) = va........................... 126 7.2.4 Complementary edges . 129 7.2.5 On the number of pseudo-similar edges . 131 7.3 Automorphism groups of the cards . 132 8 Reconstruction of random graphs 138 8.1 Random graph theory . 138 8.2 Almost all graphs are reconstructible . 140 9 Conclusion 146 9.1 Minor reconstruction . 146 9.2 Open problems in contraction reconstruction . 148 CONTENTS vii A Reconstructing simple graphs with few vertices 152 Bibliography 156 List of Symbols 157 Index 160 List of Figures 1.1 An example of a contraction deck . .2 1.2 Graphs that are not contraction reconstructible . .3 1.3 Simple graphs that are not contraction reconstructible . .3 2.1 An example of a vertex deck . .8 2.2 Two graphs that are not vertex reconstructible . .8 2.3 Graphs that are not edge reconstructible . 10 2.4 Example of deletion pseudo-similar edges . 14 2.5 Graphs that are not switching reconstructible . 16 3.1 Two graphs whose number of vertices are not reconstructible . 18 4.1 Two simple graphs that are not contraction reconstructible . 35 4.2 Trees with identical isomorphically distinct decks. 37 4.3 Trees with four edges are reconstructible . 39 4.4 Trees with five edges are reconstructible .
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