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Automorphism Groups, Isomorphism, Reconstruction (Chapter 27 of the Handbook of Combinatorics)

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Automorphism Groups, Isomorphism, Reconstruction (Chapter 27 of the Handbook of Combinatorics) Automorphism groups, isomorphism, reconstruction (Chapter 27 of the Handbook of Combinatorics) L´aszl´oBabai∗ E¨otv¨os University, Budapest, and The University of Chicago June 12, 1994 Contents 0 Introduction 3 0.1 Graphsandgroups ............................... 3 0.2 Isomorphisms, categories, reconstruction . ......... 4 1 Definitions, examples 5 1.1 Measuresofsymmetry ............................. 5 1.2 Reconstructionfromlinegraphs . .... 8 1.3 Automorphism groups: reduction to 3-connected graphs . .......... 10 1.4 Automorphismgroupsofplanargraphs . .... 11 1.5 Matrix representation. Eigenvalue multiplicity . ........... 12 1.6 Asymmetry, rigidity. Almost all graphs. Unlabelled counting ........ 14 2 Graph products 16 2.1 Prime factorization, automorphism group . ....... 17 2.2 TheCartesianproduct ............................. 18 2.3 The categorical product; cancellation laws . ........ 18 2.4 Strongproduct ................................. 19 2.5 Lexicographicproduct . .. .. 19 3 Cayley graphs and vertex-transitive graphs 20 3.1 Definition,symmetry .............................. 20 3.2 Symmetryandconnectivity . 22 3.3 Matchings, independent sets, long cycles . ....... 23 3.4 Subgraphs,chromaticnumber . .. 26 ∗Section 2 was written in collaboration with Wilfried Imrich 1 3.5 Neighborhoods, clumps, Gallai–Aschbacher decomposition ......... 27 3.6 Rateofgrowth ................................. 29 3.7 Ends....................................... 32 3.8 Isoperimetry, random walks, diameter . ...... 33 3.9 Automorphismsofmaps ............................ 38 Table: Vertex-transitive plane tilings . .... 41 3.10 Embeddingsonsurfaces,minors . .... 42 3.11 Combinatorialgrouptheory . ... 46 3.12Eigenvalues ................................... 47 4 The representation problem 48 4.1 Abstract representation; prescribed properties . ........... 48 4.2 Topologicalproperties . .. 50 4.3 Smallgraphswithgivengroup. .. 51 4.4 The concrete representation problem, 2-closure . .......... 52 5 High symmetry 54 5.1 Locally s-arc-transitivegraphs. 54 5.2 Distance-transitivegraphs . .... 55 5.3 Homogeneity .................................. 58 6 Graph isomorphism 61 6.1 Complexitytheoreticremarks . ... 61 6.2 Algorithmic results: summary of worst case bounds . ........ 61 6.3 Canonicalforms................................. 62 6.4 Combinatorial heuristics: success and failure . .......... 63 6.5 Reductions, isomorphism complete problems, Luks equivalence class . 64 6.6 Groups with restricted composition factors . ........ 65 6.7 Basicpermutationgroupalgorithms. ..... 66 6.8 Complexity of related problems . ... 68 7 The reconstruction problem 70 7.1 Vertexreconstruction. .. 70 7.2 Edgereconstruction.............................. 71 References 74 Note. While this chapter contains a substantial amount of material on infinite graphs, its focus is on finite graphs. Therefore all graphs will be finite, unless otherwise stated. Exceptions are Sections 3.6, 3.7, and 3.11, where graphs are generally infinite, and Sections 3.9, 3.10, 5.3, where a main theme is the interplay between finite and infinite. Surveys. A portion of the material discussed in this chapter is covered in two survey arti- cles on automorphism groups of graphs: Cameron [Cam83] and Babai–Goodman [BG93]. Chapter 12 of Lov´asz [Lov79a] is a nice introduction to the subject. A beautiful treatment of the basics of higher symmetry is Biggs [Big74]. Brouwer, Cohen, Neumaier [BCN89] is a 2 monumental yet enjoyable work on distance-transitivity and related subjects with detailed up-to-date information. Much of our current knowledge on graph isomorphism testing is summarized in Babai, Luks [BL83]. The general concept of reconstruction (invertibility of various constructions) is illustrated in Chapter 15 of Lov´asz [Lov79a]. Recent surveys on the Kelly-Ulam graph reconstruction conjecture include Bondy [Bon91], Ellingham [Ell88] (see also [BH77]). 0 Introduction 0.1 Graphs and groups A study of graphs as geometric objects necessarily involves the study of their symmetries, described by the group of automorphisms. Indeed, there has been significant interaction between abstract group theory and the theory of graph automorphisms, leading to the construction of graphs with remarkable properties as well as to a better understanding and occasionally a construction or proof of nonexistence of certain finite simple groups. On the other hand, in contrast to classical geometries, most finite graphs have no automor- phisms other than the identity (asymmetric graphs), a fact that is largely and somewhat paradoxically responsible for its seeming opposite: every (finite) group is isomorphic to the automorphism group of a (finite) graph. The study of graphs via their symmetries is rooted in the classical paradigm, stated in Felix Klein’s “Erlanger Programm”, that geometries are to be viewed as domains of a group action. Although graphs, as incidence structures, may seem to be degenerate geometries, we note that any incidence structure (such as a projective plane) can be represented by a graph. (The Levi graph L of an incidence structure S is a bipartite graph; its vertices correspond to the points and lines of S; and adjacency of vertices of L corresponds to point-line incidence in S.) Such representations preserve symmetry and allow fruitful generalizations (such as “generalized polygons”, Chap. 13, Sec. 7). In this chapter we try to illustrate the variety of ways in which groups and graphs inter- act. The effect of powerful results of group theory (such as the Feit–Thompson theorem on the solvability of groups of odd order) will be evident already in the introductory Sec. 1.1. Consequences of the Classification of Finite Simple Groups (CFSG) are required for some of the results in Sec. 4.3 and for the analysis of some of the algorithms in Sections 6.6, 6.7. Many of the results surveyed in Sec. 5 critically depend on the CFSG. On the other hand, some results of graph theoretic nature have played a role in the classification theory itself, as illustrated in Sections 3.5 and 5.1. In spite of these connections, the treatment of the subject will mostly be kept on an elementary level, requiring little more than basic group theory. The main theme of Sec. 3 is the surprisingly strong effect of modest symmetry assumptions on the combinatorial parameters of a graph. We try also to illustrate some of the links of the subject to areas not immediately seen to relate to groups. Sections 1.6, 7.2 illustrate this point within combinatorics. Several connections to topology are explored in Section 3 (see esp. Sections 3.6 and 3.7). Random walks feature in Sec. 3.8; linear algebra is visited briefly in Sections 1.5, 3.8, 3.12, 7.2. 3 Strong links have been forged to model theory (Sec. 5.3) and to the theory of algorithms (Sec. 6.6, 6.7). Some of the remote sources of motivation include algebraic topology (Sec. 3.11), differential geometry (Secs. 3.6, 3.7), and even a classical discovery in quantum mechanics (Sec. 1.5). 0.2 Isomorphisms, categories, reconstruction Isomorphisms of graphs are bijections of the vertex sets preserving adjacency as well as non-adjacency. In the case of directed graphs, orientations must be preserved; in the case of graphs with colored edges and/or vertices, we agree that colors, too, must be preserved. Similar definitions apply to hypergraphs. In the case of incidence structures consisting of “points” and “lines”, linked by incidence relations, we think of an isomorphism as a pair of bijections (one between the points, another between the lines), so that the pair preserves incidence. This view should be applied to graphs as well if multiple edges are allowed. Automorphisms of the graph X = (V, E) are X X isomorphisms; they form the subgroup Aut(X) of the symmetric group Sym(V ). Automorphisms→ of directed graphs, etc., are defined analogously. The questions of reconstruction are, broadly speaking, questions of invertibility of cer- tain isomorphism preserving operations on structures. A category in which all morphisms are isomorphisms is called a Brandt groupoid. Let , be two Brandt groupoids and C D F : a functor. Hence X = Y implies F (X) = F (Y ). We call F weakly re- C →D ∼ ∼ constructible if the converse also holds: F (X) ∼= F (Y ) implies X ∼= Y . We say that F is strongly reconstructible if for every pair X,Y of objects of , F induces a bijec- tion between the sets Iso(X,Y ) and Iso(F (X), F (Y )) of isomorphisms.C In this case, Aut(X) ∼= Aut(F (X)) for every object X. We also say that, within the class , the object X is (weakly, strongly) reconstructible from F (X). C A classical example is the reconstructibility of a multiset of direct irreducible finite groups from their direct product (unique direct factorization, R. Remak – O. Yu. Schmidt, cf. Baer [Bae47]). The category consists of the multisets of direct irreducible finite groups C with the natural notion of isomorphism. Let F associate the direct product of the members of such a multiset X with X. This functor is weakly but not strongly reconstructible. (To see the latter, consider the pair Zp, Zp .) Homomorphisms of graphs{ are defined} as adjacency preserving maps, i.e., a map f : V V is a homomorphism of the graph X =(V , E ) to the graph X =(V , E ) if 1 → 2 1 1 1 2 2 2 (f(x), f(y)) E2 whenever (x, y) E1. It is not required that nonadjacency be preserved; therefore a bijective∈ homomorphism∈ is not necessarily an isomorphism. It is easy to see that the chromatic number of the graph X is the smallest (cardinal) number m such that the set Hom(X,Km) of X
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