منابع مشابه
The metric dimension and girth of graphs
A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...
متن کاملNote on Metric Dimension
The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension ...
متن کاملMetric Dimension for Random Graphs
The metric dimension of a graph G is the minimum number of vertices in a subset S of the vertex set of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension of the random graph G(n, p) for a wide range of probabilities p = p(n).
متن کاملValiant Metric Embeddings , Dimension Reduction
In the previous lecture notes, we saw that any metric (X, d) with |X| = n can be embedded into R 2 n) under any the `1 metric (actually, the same embedding works for any `p metic), with distortion O(log n). Here, we describe an extremely useful approach for reducing the dimensionality of a Euclidean (`2) metric, while incurring very little distortion. Such dimension reduction is useful for a nu...
متن کاملthe metric dimension and girth of graphs
a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...
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ژورنال
عنوان ژورنال: Scholarpedia
سال: 2019
ISSN: 1941-6016
DOI: 10.4249/scholarpedia.53881