On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

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

Cellular and ultrastructural characterization of the grey-morph phenotype in southern right whales (Eubalaena australis)

Southern right whales (SRWs, Eubalena australis) are polymorphic for an X-linked pigmentation pattern known as grey morphism. Most SRWs have completely black skin with white patches on their bellies and occasionally on their backs; these patches remain white as the whale ages. Grey morphs (previously referred to as partial albinos) appear mostly white at birth, with a splattering of rounded black marks; but as the whales age, the white skin gradually changes to a brownish grey color. The cellular and developmental bases of grey morphism are not understood. Here we describe cellular and ultrastructural features of grey-morph skin in relation to that of normal, wild-type skin. Melanocytes were identified histologically and counted, and melanosomes were measured using transmission electron microscopy. Grey-morph skin had fewer melanocytes when compared to wild-type skin, suggesting reduced melanocyte survival, migration, or proliferation in these whales. Grey-morph melanocytes had smaller melanosomes relative to wild-type skin, normal transport of melanosomes to surrounding keratinocytes, and normal localization of melanin granules above the keratinocyte nuclei. These findings indicate that SRW grey-morph pigmentation patterns are caused by reduced numbers of melanocytes in the skin, as well as by reduced amounts of melanin production and/or reduced sizes of mature melanosomes. Grey morphism is distinct from piebaldism and albinism found in other species, which are genetic pigmentation conditions resulting from the local absence of melanocytes, or the inability to synthesize melanin, respectively