id: 06099133
dt: j
an: 06099133
au: Jannesari, Mohsen; Omoomi, Behnaz
ti: The metric dimension of the lexicographic product of graphs.
so: Discrete Math. 312, No. 22, 3349-3356 (2012).
py: 2012
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: lexicographic product; resolving set; metric dimension; metric basis;
    adjacency dimension
ci: 
li: doi:10.1016/j.disc.2012.07.025
ab: Summary: For a set $W$ of vertices and a vertex $v$ in a connected graph
    $G$, the $k$-vector $r_{W}(v)=(d(v,w_{1}),\ldots ,d(v,w_{k}))$ is the
    metric representation of $v$ with respect to $W$, where
    $W=\{w_{1},\ldots ,w_{k}\}$ and $d(x,y)$ is the distance between the
    vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if
    distinct vertices of $G$ have distinct metric representations with
    respect to $W$. The minimum cardinality of a resolving set for $G$ is
    its metric dimension. In this paper, we study the metric dimension of
    the lexicographic product of graphs $G$ and $H$, denoted by $G[H]$.
    First, we introduce a new parameter, the adjacency dimension, of a
    graph. Then we obtain the metric dimension of $G[H]$ in terms of the
    order of $G$ and the adjacency dimension of $H$.
rv: