Thinness of product graphs

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.Comment: 45 page

Coexistence of competing first passage percolation on hyperbolic graphs

We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probability. We show that this is the case when $G$ is vertex transitive, non-amenable and hyperbolic, in particular, for any $\lambda>0$ there is a $\mu_0=\mu_0(G,\lambda)>0$ such that for all $\mu\in(0,\mu_0)$ the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model. We also show that $\text{FPP}_\lambda$ produces an infinite cluster almost surely for any positive $\lambda,\mu$, establishing fundamental differences with the behavior of such processes on $\mathbb{Z}^d$.Comment: 53 pages, 13 figure