Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

Splittings of generalized Baumslag-Solitar groups

We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (`fully reduced' decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag-Solitar groups with no non-trivial integral moduli.Comment: 20 pages; hyperlinked latex. Version 2: minor change

Fixed-point free circle actions on 4-manifolds

This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) $S^1$-action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free $S^1$-actions under some further conditions on the fundamental group. The connection between the classification of the $S^1$-manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the $S^1$-manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free $S^1$-action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold. The paper ends with two questions concerning the topological nature of the smooth classification and the Seiberg-Witten invariants of 4-manifolds admitting a smooth fixed-point free $S^1$-action.Comment: 42 pages, no figures, Algebraic and Geometric Topolog

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.Comment: This replacement is part I of the final version of the paper, which has been split into two parts. The second part is available from the arXiv under the title "Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation" arXiv:2001.0872

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family