We study variational principles for metric mean dimension. First we prove that in the principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second derive a dim

In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that any pair graph, there is vertex from whose two distances to are distinct. This problem has mainly been studied on undirected graphs and gained lot attention in recent years. We focus directed graphs, show how solve linear-time digraphs underlying graph (ignoring multiple edges) tree. (nontrivially) extends ...

Diberikan graf terhubung dan sederhana $G=(V(G), E(G))$ bilangan bulat positif $k$. Himpunan $S \subseteq V(G)$ disebut sebagai pembangkit $k$-metrik jika untuk setiap pasang titik berbeda $u,v \in V(G)$, terdapat paling sedikit $k$ $w_{1}, w_{2}, \ldots, w_{k} S$ sedemikian sehingga $d(u,w_{i}) \neq d(v,w_{i})$ $i \{1,2,\ldots, k\}$, dengan $d(u,v)$ adalah panjang \emph{path} terpendek dari $u...