A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. The \emph{strong metric dimension}, $sdim(G)$, of a graph $G$ is the minimum among cardinalities of all strong resolving sets: $W \subseteq V(G)$ is a \emph{strong resolving set} of $G$ if for any $u, v \in V(G)$, there exists an $x \in W$ such that either $u$ lies on an $x-v$ geodesic or $v$ lies on an $x-u$ geodesic. In this paper, we prove that $Z(G) \le sdim(G)+3r(G)$ for a connected graph $G$, where $r(G)$ is the cycle rank of $G$. Further, we prove the sharp bound $Z(G) \leq sdim(G)$ when $G$ is a tree or a unicyclic graph, and we characterize trees $T$ attaining $Z(T)=sdim(T)$. It is easy to see that $sdim(T+e)-sdim(T)$ can be arbitrarily large for a tree $T$; we prove that $sdim(T+e) \ge sdim(T)-2$ and show that the bound is sharp.Comment: 8 pages, 5 figure

The generalized 3-edge-connectivity of lexicographic product graphs

The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly study the generalized 3-edge-connectivity of $G \circ H$, and get upper and lower bounds of $\lambda_3(G \circ H)$. Moreover, all bounds are sharp.Comment: 14 page

The first global deep-sea stable isotope assessment reveals the unique trophic ecology of Vampire Squid Vampyroteuthis infernalis (Cephalopoda)

Vampyroteuthis infernalis Chun, 1903, is a widely distributed deepwater cephalopod with unique morphology and phylogenetic position. We assessed its habitat and trophic ecology on a global scale via stable isotope analyses of a unique collection of beaks from 104 specimens from the Atlantic, Pacific and Indian Oceans. Cephalopods typically are active predators occupying a high trophic level (TL) and exhibit an ontogenetic increase in δ15N and TL. Our results, presenting the first global comparison for a deep-sea invertebrate, demonstrate that V. infernalis has an ontogenetic decrease in δ15N and TL, coupled with niche broadening. Juveniles are mobile zooplanktivores, while larger Vampyroteuthis are slow-swimming opportunistic consumers and ingest particulate organic matter. Vampyroteuthis infernalis occupies the same TL (3.0–4.3) over its global range and has a unique niche in deep-sea ecosystems. These traits have enabled the success and abundance of this relict species inhabiting the largest ecological realm on the planet.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. The attached file is the published pdf