An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can “move” to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence...

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique c...

An eternal $1$-secure set, in a graph $G = (V, E)$ is set $D \subset V$ having the property that for any finite sequence of vertices $r_1, r_2, \ldots, r_k$ there exists $v_1, v_2, v_k$ and $ D D_0, D_1, D_2, D_k$ dominating sets $G$, such each $i$, $1 \leq i k$, $D_{i} (D_{i-1} - \{v_i\}) \cup \{r_i\}$, where $v_i \in D_{i-1}$ $r_i N[v_i]$. Here v_i$ possible. The cardinality smallest $G$ call...

The automated extraction of tempo and beat information from music recordings is a challenging task. Especially in the case of expressive performances, current beat tracking approaches still have significant problems to accurately capture local tempo deviations and beat positions. In this paper, we introduce a novel evaluation framework for detecting critical passages in a piece of music that ar...

We show that for every minimum eternal dominating set, D, of a graph G and every vertex v ∈ D, there is a sequence of attacks at the vertices of G which can be defended in such a way that an eternal dominating set not containing v is reached. The study of the stronger assertion that such a set can be reached after a single attack is defended leads to the study of graphs which are critical in th...

Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow existence of points that never thermalize. I explore the fractal geometry of the resulting spacetime, using coordinate-independent quantities. The formalism of stochastic inflation can be used to obtain the fractal dimension of the set of eternally inflating points (the “eternal...