Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

Topology of Cut Complexes of Graphs

We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner (1998). We describe the effect of various graph operations on the cut complex, and study its shellability, homotopy type and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism $K_n \times K_2$, using techniques from algebraic topology, discrete Morse theory and equivariant poset topology.Comment: 36 pages, 10 figures, 1 table, Extended Abstract accepted for FPSAC2023 (Davis

The galaxy of Coxeter groups

In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.Comment: 30 pages, 6 figures. v2: Incorporated referee's suggestions; Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved other proofs and text; Final version, to appear in the Journal of Algebr

Average Betti numbers of induced subcomplexes in triangulations of manifolds

We study a variation of Bagchi and Datta’s σ-vector of a simplicial complex C, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of C. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of C. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given f-vector. For the first entry of σ, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on σ can be used to obtain lower bounds on the f-vector of triangulated 4-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.Santos is also supported by grants MTM2014-54207-P and MTM2017-83750-P of the Spanish Ministry of Science

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

A commonly studied means of parameterizing graph problems is the deletion distance from triviality [11], which counts vertices that need to be deleted from a graph to place it in some class for which e cient algorithms are known. In the context of graph isomorphism, we de ne triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is FPT when parameterized by elimination distance to bounded degree, extending results of Bouland et al.The work was supported in part by EPSRC grant EP/H026835, DAAD grant A/13/05456, and DFG project Logik, Struktur und das Graphenisomorphieproblem.This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s00453-015-0045-