THE SIMILARITY OF METRIC DIMENSION AND LOCAL METRIC DIMENSION OF ROOTED PRODUCT GRAPH

THE SIMILARITY OF METRIC DIMENSION AND LOCAL METRIC DIMENSION OF ROOTED PRODUCT GRAPH
L. Susilowati; Slamin, Slamin; M. I. Utoyo; N. Estuningsih
Let G be a connected graph with vertex set ( )GV and =W
{}()….,,,
⊂ The representation of a vertex ( )GVv ∈ with
respect to W is the ordered k-tuple
( ) ( ) ( )( …,,,,,
GVwww
21
()),,
k
k
1

wvdwvdWvr =|
21
wvd where ()wvd , represents the distance between vertices v
and w. The set W is called a resolving set for G if every vertex of G
has a distinct representation. A resolving set containing a minimum number of vertices is called basis for G. The metric dimension of G,
denoted by
(),dim G is the number of vertices in a basis of G. If every
two adjacent vertices of G have a distinct representation with respect
to W, then the set W is called a local resolving set for G and the
minimum local resolving set is called a local basis of G. The
cardinality of a local basis of G is called local metric dimension
of G, denoted by
( ).dim G
l

In this paper, we study the local metric
dimension of rooted product graph and the similarity of metric
dimension and local metric dimension of rooted product graph.
Far East Journal of Mathematical Sciences (FJMS), Volume 97, Number 7, 2015, Pages 841-856
Source: http://repository.unej.ac.id/

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