Automorphism groups of graphs and edge-contraction

AbstractIf a class C of finite graphs is closed under contraction and forming subgraphs, and if every finite abstract group occurs as the automorphism group of some graph in C, then C contains all finite graphs (up to isomorphism). Also related results concerning automorphism groups of graphs on given surfaces are mentioned

Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach

It can be shown that each permutation group G ? ?_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, ?_n itself can be embedded in the n-clique K_n, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group G? ?_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G ? ?_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree ?, can also be generated by a context-free grammar of size 2^{O(k?log?)}? m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(k?log?)}? m^{O(k)}. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{?(n)} lower bound on the grammar complexity of the symmetric group ?_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of ?_n of index 2^{cn} for some small constant c must have n^{?(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{?} for ? < 1 and maximum degree o(n/log n)

On endomorphism universality of sparse graph classes

Solving a problem of Babai and Pultr from 1980 we show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a graph of bounded degree. Indeed we show that maximum degree $3$ suffices, which is best-possible. On the way we generalize a classic result of Frucht by showing that every group is the endomorphism monoid of a graph of maximum degree $3$ and we answer a question of Ne\v{s}et\v{r}il and Ossona de Mendez from 2012, presenting a class of bounded expansion such that every monoid is the endomorphism monoid of one of its members. On the other hand we strengthen a result of Babai and Pultr and show that no class excluding a topological minor can have all completely regular monoids among its endomorphism monoids. Moreover, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids.Comment: 37 pages, 18 figure

Combinatorial Properties of Finite Models

We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well known structures that are finitely presented. We focus on structures endowed with natural partial orders and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism orders for various combinatorial objects. We give a new combinatorial proof of the existence of embedding-universal objects for homomorphism-defined classes of structures. This relates countable embedding-universal structures to homomorphism dualities (finite homomorphism-universal structures) and Urysohn metric spaces. Our explicit construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font