Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic and heteroclinic orbits of saddle points. These orbits are most readily computed and studied as intersections of unstable and stable manifolds comprising homoclinic or heteroclinic tangles in the surface. We show how to compute a map of a one-dimensional space similar to a train-track which represents the isotopy-stable dynamics of the surface diffeomorphism relative to a tangle. All orbits of this one-dimensional representative are globally shadowed by orbits of the surface diffeomorphism, and periodic, homoclinic and heteroclinic orbits of the one-dimensional representative are shadowed by similar orbits in the surface.By constructing suitable surface diffeomorphisms, we prove that these results are optimal in the sense that the topological entropy of the one-dimensional representative is the greatest lower bound for the entropies of diffeomorphisms in the isotopy class.Comment: Version submitted to "Dynamical Systems: An International Journal" Section 7 has been further revised; the method for pA maps is new. Notation has been standardised throughou

The genus of curve, pants and flip graphs

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface. The lower bounds are shown using probabilistic methods.Comment: 26 pages, 9 figure

Exact Localisations of Feedback Sets

The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph $G=(V,E)$ into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ of all elementary cycles or simple cycles running through some arc $e \in E$, can be computed in $\mathcal{O}\big(|E|^2\big)$ and $\mathcal{O}(|E|^4)$, respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in $\mathcal{O}(|V||E|^3)$. We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times $\mathcal{O}\big(2^{m}|E|^4\log(|V|)\big)$ and $\mathcal{O}\big(2^{n}\Delta(G)^4|V|^4\log(|E|)\big)$, where $m \leq |E|-|V| +1$, $n \leq (\Delta(G)-1)|V|-|E| +1$, respectively. The parameters $m,n$ can be computed in $\mathcal{O}(|E|^3)$, $\mathcal{O}(\Delta(G)^3|V|^3)$, respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, $m,n$ equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER

On retracts, absolute retracts, and folds in cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

Quantifying the Extent of Lateral Gene Transfer Required to Avert a `Genome of Eden'

The complex pattern of presence and absence of many genes across different species provides tantalising clues as to how genes evolved through the processes of gene genesis, gene loss and lateral gene transfer (LGT). The extent of LGT, particularly in prokaryotes, and its implications for creating a `network of life' rather than a `tree of life' is controversial. In this paper, we formally model the problem of quantifying LGT, and provide exact mathematical bounds, and new computational results. In particular, we investigate the computational complexity of quantifying the extent of LGT under the simple models of gene genesis, loss and transfer on which a recent heuristic analysis of biological data relied. Our approach takes advantage of a relationship between LGT optimization and graph-theoretical concepts such as tree width and network flow

Equational reasoning with context-free families of string diagrams

String diagrams provide an intuitive language for expressing networks of interacting processes graphically. A discrete representation of string diagrams, called string graphs, allows for mechanised equational reasoning by double-pushout rewriting. However, one often wishes to express not just single equations, but entire families of equations between diagrams of arbitrary size. To do this we define a class of context-free grammars, called B-ESG grammars, that are suitable for defining entire families of string graphs, and crucially, of string graph rewrite rules. We show that the language-membership and match-enumeration problems are decidable for these grammars, and hence that there is an algorithm for rewriting string graphs according to B-ESG rewrite patterns. We also show that it is possible to reason at the level of grammars by providing a simple method for transforming a grammar by string graph rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21145-9_