Related papers: Domination Between Trees and Application to an Exp…
Let $\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. In 1977, Hedetniemi and Mitchell began with the comparison of of $i(G)$ and $\gamma(G)$ and recently Rad and Volkmann posted a…
We consider random dynamics on a uniform random recursive tree with $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in…
We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…
Dispersal is an important strategy that allows organisms to locate and exploit favorable habitats. The question arises: given competition in a spatially heterogeneous landscape, what is the optimal rate of dispersal? Continuous population…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…
We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate…
Let $T$ be a regular rooted tree. For every natural number $n$, let $B_n$ be the finite subtree of vertices with graph distance at most $n$ from the root. Consider the following forest-fire model on $B_n$: Each vertex can be "vacant" or…
An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability…
Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics…
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the…
Dominating sets in graphs are often used to model some monitoring of the graph: guards are posted on the vertices of the dominating set, and they can thus react to attacks occurring on the unguarded vertices by moving there (yielding a new…
In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must…
We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of…
We study the problem of comparing ageing patterns of the lifetime of k-out-of-n systems. Mathematically, this reduces to being able to decide about a stochastic ordering relationship between different order statistics. We discuss such…
The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove…
We consider a class of biologically-motivated stochastic processes in which a unicellular organism divides its resources (volume or damaged proteins, in particular) symmetrically or asymmetrically between its progeny. Assuming the final…
The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and…
Stochastic dominance is an important concept in probability theory, econometrics and social choice theory for robustly modeling agents' preferences between random outcomes. While many works have been dedicated to the univariate case, little…
The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown…