aa r X i v : . [ m a t h . C O ] F e b Reconstruction of small graphs and tournaments
Brendan D. McKay
School of ComputingAustralian National UniversityCanberra, ACT 2601, Australia [email protected]
VERSION 1: 27 Jan 2021
Abstract
We describe computer searches that prove the graph reconstruction conjecturefor graphs with up to 13 vertices and some limited classes on larger sizes. We alsoinvestigate the reconstructability of tournaments up to 13 vertices and posets up to13 points. In all cases, our proofs also apply to the set reconstruction problem thatuses the isomorph-reduced deck. s:intro
The late Frank Harary visited the University of Melbourne in 1976, when I was a Mas-ters student there. I mentioned to him that I had proved the 9-point graphs to bereconstructible, using a catalogue of graphs obtained on magnetic tape from Canada [2].Harary replied, “Send it to the Journal of Graph Theory. I accept it!”. Impressed byhaving a paper I hadn’t written yet accepted for a journal that hadn’t started publishingyet, I quickly wrote it up and did as Harary suggested [11].We recount the standard definitions. Except when we say otherwise, our graphs aresimple, undirected and labelled. We use the standard sloppy terminology that an un-labelled graph is the isomorphism type of a graph, or a labelled graph with the labelshidden. For an unlabelled graph G with vertex v , the card G − v is the unlabelled graphobtained from G by removing v . The deck of G is the multiset of its cards. The celebrated Kelly-Ulam reconstruction conjecture says that G can be uniquely determined fromits deck if G has at least 3 vertices. 1e will focus on a stronger version of the conjecture due to Harary [5], since it is nomore onerous for the computer. Define ID ( G ), called the reduced deck of G , to be the set of cards of G . That is, ID ( G ) tells us which unlabelled graphs appear in the deck,but not how many of each there are. The set reconstruction conjecture is that G isuniquely determined by ID ( G ) if it has at least 4 vertices.Many surveys of the reconstruction conjecture have been written; see Lauri [8] for afairly recent one.Twenty years after investigating the 9-point graphs, the author extended the search to11 vertices [12]. Since the number of graphs on 11 vertices is 1,018,997,864, a completelynew computational method was required. Now more than another twenty years havepast, so it is time for a further extension and that is the purpose of this project. Despitehaving more and faster computers, it is sobering that the number of graphs on 13 verticesis 50,502,031,367,952. Our method will be similar to [12] but with some improvementsto make the task less onerous. The weaker edge-reconstruction conjecture, which weotherwise will not consider, was meanwhile checked up to 12 vertices by Stolee [20]. s:Graphs If X is a structure built from { , , . . . , n } , and φ ∈ S n where S n is the symmetric groupon { , , . . . , n } , then X φ is obtained from X by replacing each i by i φ . For example, if G is a graph with vertices { , , . . . , n } , then the graph G φ has an edge i φ j φ for each edge ij of G . The automorphism group of G is Aut( G ) = { φ ∈ S n | G φ = G } where “=” denotesequality not isomorphism.Let C be a non-empty class of labelled graphs that is closed under isomorphism andtaking induced subgraphs, and let C n be the subset of C containing those with n vertices.Clearly C = { K } . We will assume that the vertices of G ∈ C n are { , , . . . , n } . Thespecial case that C contains all graphs will be denoted by G and similarly G n .For G ∈ G and W ⊆ V ( G ), let G [ W ] v denote the graph obtained from G by appendinga new vertex v and joining it to each of the vertices in W . Define (cid:22) to be a preorder(reflexive, transitive order) on labelled graphs, invariant under isomorphism. (An examplewould that G (cid:22) G iff G has at most as many edges as G .) Consider the followingpossible properties of a function m : G → N .(A) For each H ∈ G , m ( H ) is an orbit of Aut( H ).(A ′ ) For each H ∈ G , m ( H ) is the union of all the orbits of Aut( H ) such that the cards H − v for v ∈ m ( H ) are maximal under (cid:22) amongst all cards of H .(B) For each H ∈ G n and φ ∈ S n , m ( H φ ) = m ( H ) φ .2 lgorithm generate ( G : labelled graph; n : integer) if | V ( G ) | = n thenoutput G elsefor each orbit A of the action of Aut( G ) on 2 V ( G ) do select any W ∈ A and form H := G [ W ] v if H ∈ G and v ∈ m ( H ) then generate ( H, n ) endifendforendifend generate Figure 1: Generation algorithm fig:generate
Now consider the algorithm shown in Figure 1. When we form H := G [ W ] v in theinner loop, we say that G is a parent of H and H is a child of G .We have the following theorem. generate Theorem 2.1.
Let C be a non-empty class of labelled graphs that is closed under isomor-phism and taking induced subgraphs. Then:(a) If m : G → N satisfies (A) and (B), then calling generate ( K , n ) will cause outputof exactly one member of each isomorphism class of C n .(b) Suppose m : G → N satisfies (A) and (B) for | V ( H ) | < n and (A ′ ) for | V ( H ) | = n .Let G , G be non-isomorphic members of C n with the same reduced deck. Thencalling generate ( K , n ) will cause G and G to be output as children of the samenon-empty set of parents.Proof. Part (a) is proved in [13]. This is the canonical construction path method whichhas been widely adopted for isomorph-free generation.In part (b), it is no longer true that G and G will be (up to isomorphism) outputonly once. However, as we will show, both will be output at least once, and from thesame set of parents. Let G − v be a card of G maximal under (cid:22) . Since G has the samereduced deck as G , there is a card G − w maximal under (cid:22) and isomorphic to G − v . Bypart (a), the call generate ( G ′ , n −
1) is made for some G ′ isomorphic to G − v and G − w .During that call we construct (up to to isomorphism) all 1-vertex extensions of G ′ that liein C n , so in particular some H = G ′ [ W ] v isomorphic to G and H = G ′ [ W ] w isomorphicto G . Since they pass the tests v ∈ m ( H ) and w ∈ m ( H ), the calls generate ( H , n )and generate ( H , n ) are both made, causing H and H to be output.3he great advantage of this method is that most parents only have a small numberof children even if the total number of graphs is huge. So detailed comparison of reduceddecks can be carried out in small batches without the need to store many graphs at once.Our code is based on the implementation geng of algorithm generate in the author’spackage nauty [14]. For (cid:22) we used a hash code based on the number of edges and trianglesin the cards. For large sizes, most graphs have trivial automorphism groups and the hashcode distinguishes between cards quite well, so the total number of graphs constructed isnot much greater than the number of isomorphism classes. After collecting the children ofeach parent, we compute an invariant of the reduced decks based on the degree sequencesof the cards, and then reject any child which is unique. For those remaining, we doa complete isomorphism check of the cards with the most edges, and for any still notdistinguished a complete isomorphism check of all the cards.As an example, there are 1,018,997,864 graphs with 11 vertices. The testing programmade 1,131,624,582, an increase of only 11%. The time for testing was only 2.4 times thegeneration time.Theorem 2.1 refers to detection of non-reconstructible graphs within a class C , so itis important to know when membership of the class is determined by the reduced deck.This eliminates the possibility that a graph in C has the same reduced deck as a graphnot in C . properties Lemma 2.2.
Let G and G be graphs on n ≥ vertices with the same reduced decks.Then the following are true.(a) G and G have the same minimum and maximum degrees.(b) For ≤ k < n , either both or neither G and G contain a k -cycle.(c) Either both or neither G and G are bipartite.Proof. Part (a) was proved by Manvel [9, 10]. Part (b) is obvious as the cycles of lengthless than n are those appearing in the cards. For part (c), note that a non-bipartite graph G with n vertices either has an odd cycle of length less than n or G is an n -cycle. Thelatter situation is uniquely characterised by the reduced deck being a single path. s:Resultsgraphs Theorem 3.1.
For at least 4 vertices, all graphs in the following classes are recon-structible from their reduced decks (and therefore reconstructible).(a) Graphs with at most 13 vertices.(b) Graphs with no triangles and at most 16 vertices.(c) Graphs of girth at least 5 and at most 20 vertices. d) Graphs with no -cycles and at most 19 vertices.(e) Bipartite graphs with at most 17 vertices.(f ) Bipartite graphs of girth at least 6 and at most 24 vertices.(g) Graphs with maximum degree at most 3 and at most 22 vertices.(h) Graphs with degrees in the range [ δ, ∆ ] and at most n vertices, where ( δ, ∆ ; n ) is (0 ,
5; 14) , (5 ,
6; 14) , (6 ,
7; 14) , (0 ,
4; 15) , (4 ,
5; 15) or (3 ,
4; 16) . This theorem required testing of more than 6 × graphs and took about 1.5 yearson Intel cpus running at approximately 3 GHz. s:Tournaments The reconstruction problem is defined for tournaments in the same way as for graphs,but in this case many counterexamples are known. Harary and Palmer [6] stated theproblem and gave counterexamples with 3 or 4 vertices, while Beineke and Parker [3] gavecounterexamples with 5 and 6 vertices. Beineke and Parker thought they had searched 6vertices exhaustively (by hand) but they missed one counterexample. That extra case,together with the absence of any counterexamples on 7 vertices and two counterexampleson 8 vertices were found by Stockmeyer in 1975 [16].The most important breakthrough was made by Stockmeyer [17,18], see also Kocay [7],who showed that there is a non-reconstructible pair of tournaments on any order of theform 2 i + 2 j for 0 ≤ i < j .A complete search up to 12 vertices was carried out by Abrosimov and Dolgov [1]. Theyfound only Stockmeyer’s counterexamples for 10 and 12 vertices, and no counterexamplesfor 11 vertices. Their method was similar to ours but took much longer due to beingbased on a far slower generation algorithm. Kocay independently checked these resultsup to 10 vertices in 2018 (unpublished).In summary, for 3–12 vertices, the number of pairs of non-reconstructible tournamentsis respectively 1, 1, 1, 4, 0, 2, 1, 1, 0, 1.To the best of our knowledge, there is no literature on the set reconstruction problemfor tournaments. Using methods similar to Section 2, we have found all tournaments upto 13 vertices which are not determined by their reduced decks. tournaments Theorem 4.1.
The tournaments with at most 13 vertices which are not determined upto isomorphism by their reduced decks are:(a) The known tournaments not reconstructible from their (full) decks, as above. fig:tournaments (b) A tournament with 4 vertices that has the same reduced deck as a pair of tournamentswith the same deck.(c) Two tournaments with 5 vertices having the same reduced deck.In particular, all tournaments with 13 vertices are determined by their reduced decks. Thetournaments mentioned in (b) and (c) are shown in Figure 2.
This theorem required testing of about 5 × tournaments and took about 2 yearson Intel cpus at approximately 3 GHz. Graph reconstruction problems are special cases of reconstruction problems for binaryrelations. See Rampon [15] for a survey. In this paper, the only non-graph reconstructionproblem we will mention is for partially-ordered sets (posets). Note that removing a pointfrom a poset is the same as removing a vertex from the corresponding transitive digraph,but not the same as removing a vertex from the Hasse diagram.The reader can check that all of the posets on 2 points, and three of the five posetson 3 points, have the same reduced deck. For 4–13 points, we have checked that everyposet has a unique reduced deck (amongst posets). The number of posets with 13 pointsis 33,823,827,452. This was a quick computation of about 2 weeks that used the generatordescribed in [4] to make all the reduced decks directly. The method of Section 2 wouldenable the computation to continue up to 15 points, but we leave this for another timeand another place (those who knew Paul Erd˝os will understand this allusion).6
Acknowledgement
This research/project was undertaken with the assistance of resources and services fromthe National Computational Infrastructure (NCI), which is supported by the AustralianGovernment.
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