The connected circular metric dimension of a graph

Let be a simple graph and be any two vertices of G. Then the circular
distance between and denoted by and is defined by
{
where and are detour distance and distance between and respectively. Let
and . The representation of with respect
to is the -tuple ( ) Then is called a circular resolving
set if different vertices of have different representations with respect to . A circular resolving
set W is called connected circular resolving set, is connected . The minimum cardinality of
a connected circular resolving set in a graph G is its connected circular metric dimension of G and
is denoted by . The connected circular metric dimension of some standard graphs are
determined. Some general properties satisfies by this concept are studied. Connected graphs of
order with connected circular metric dimension 1 are characterized. Necessary condition for
the connected circular metric dimension to be is given.