Enumeration of 22-Polymatroids on up to Seven Elements

A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of $2$-polymatroids, up to isomorphism. When we implemented this algorithm on a computer, obtaining all $2$-polymatroids on at most seven elements, we discovered the surprising fact that the number of $2$-polymatroids on seven elements fails to be unimodal in rank.Comment: 9 page

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

Isomorphism testing of kk-spanning tournaments is Fixed Parameter Tractable

An arc-colored tournament is said to be $k$-spanning for an integer $k\geq 1$ if the union of its arc-color classes of maximal valency at most $k$ is the arc set of a strongly connected digraph. It is proved that isomorphism testing of $k$-spanning tournaments is fixed-parameter tractable.Comment: 8 page

On the automorphism groups of strongly regular graphs II

We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp((logn)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ákos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem

Hypergraph Isomorphism for Groups with Restricted Composition Factors

We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group ? over domain V, and asking whether there is a permutation ? ? ? that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1). As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c)

Graph Isomorphism for unit square graphs

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.Comment: 31 pages, 6 figure

Every Property of Hyperfinite Graphs is Testable

A k-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most k from v. We show that the structure of a planar graph on n vertices, and with constant maximum degree d, is determined, up to the modification (insertion or deletion) of at most εdn edges, by the frequency of k-discs for certain k=k(ε,d) that is independent of the size of the graph. We can replace planar graphs by any hyperfinite class of graphs, which includes, for example, every graph class that does not contain a set of forbidden minors. A pure combinatorial consequence of this result is that two d-bounded degree graphs that have similar frequency vectors (that is, the l_1 difference between the frequency vectors is small) are close to isomorphic (where close here means that by inserting or deleting not too many edges in one of them, it becomes isomorphic to the other). We also obtain the following new results in the area of property testing, which are essentially equivalent to the above statement. We prove that (a) graph isomorphism is testable for every class of hyperfinite graphs, (b) every graph property is testable for every class of hyperfinite graphs, (c) every hyperfinite graph property is testable in the bounded degree graph model, (d) A large class of graph parameters is approximable for hyperfinite graphs. Our results also give a partial explanation of the success of motifs in the analysis of complex networks