Stallings graphs for quasi-convex subgroups

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the conjugacy or the almost malnormality problems. Our results extend earlier algorithmic results for the more restricted class of virtually free groups. We also extend our construction to relatively quasi-convex subgroups of relatively hyperbolic groups, under certain additional conditions.Comment: 40 pages. New and improved versio

On several extremal problems in graph theory involving gromov hyperbolicity constant

Mención Internacional en el título de doctorIn this Thesis we study the extremal problems of maximazing and minimazing the hyperbolicity constant on several families of graphs. In order to properly raise our research problem, we need to introduce some important definitions and make some remarks on the graphs we study.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Elena Romera Colmenarejo.- Secretario: Ana María Portilla Ferreira.- Vocal: José María Sigarreta Almir

Maximizing the minimum and maximum forcing numbers of perfect matchings of graphs

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for bipartite graphs $G$, $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. In this paper, we solve the problem for general graphs and obtain that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or $K^+_{n,n}$ ($K_{n,n}$ with arbitrary additional edges in the same partite set). For a larger class of graphs $G$ with $F(G)=n-1$ we show that $G$ is $n$-connected and a brick (3-connected and bicritical graph) except for $K^+_{n,n}$. In particular, we prove that the forcing spectrum of each such graph $G$ is continued by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $\lfloor\frac{n}{2}\rfloor$ to $n-1$

Heat kernel for reflected diffusion and extension property on uniform domains

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math. 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Ast\'erisque 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (PTRF 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpi\'nski carpet equipped with the self-similar Dirichlet form.Comment: 56 pages; comments welcom

Quarc: an architecture for efficient on-chip communication

The exponential downscaling of the feature size has enforced a paradigm shift from computation-based design to communication-based design in system on chip development. Buses, the traditional communication architecture in systems on chip, are incapable of addressing the increasing bandwidth requirements of future large systems. Networks on chip have emerged as an interconnection architecture offering unique solutions to the technological and design issues related to communication in future systems on chip. The transition from buses as a shared medium to networks on chip as a segmented medium has given rise to new challenges in system on chip realm. By leveraging the shared nature of the communication medium, buses have been highly efficient in delivering multicast communication. The segmented nature of networks, however, inhibits the multicast messages to be delivered as efficiently by networks on chip. Relying on extensive research on multicast communication in parallel computers, several network on chip architectures have offered mechanisms to perform the operation, while conforming to resource constraints of the network on chip paradigm. Multicast communication in majority of these networks on chip is implemented by establishing a connection between source and all multicast destinations before the message transmission commences. Establishing the connections incurs an overhead and, therefore, is not desirable; in particular in latency sensitive services such as cache coherence. To address high performance multicast communication, this research presents Quarc, a novel network on chip architecture. The Quarc architecture targets an area-efficient, low power, high performance implementation. The thesis covers a detailed representation of the building blocks of the architecture, including topology, router and network interface. The cost and performance comparison of the Quarc architecture against other network on chip architectures reveals that the Quarc architecture is a highly efficient architecture. Moreover, the thesis introduces novel performance models of complex traffic patterns, including multicast and quality of service-aware communication