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Let $\Gamma$ be a simple connected undirected graph with vertex set$V(\Gamma)$ and edge set $E(\Gamma)$. The metric dimension of a graph $\Gamma$is the least number of vertices in a set with the property that the list ofdistances from any vertex to those in the set uniquely identifies that vertex.For an ordered subset $W = \{w_1, w_2, ..., w_k\}$ of vertices in a graph$\Gamma$ and a vertex $v$ of $\Gamma$, the metric representation of $v$ withrespect to $W$ is the $k$-vector $r(v | W) = (d(v, w_1), d(v, w_2), ..., d(v,w_k ))$. If every pair of distinct vertices of $\Gamma$ have different metricrepresentations then the ordered set $W$ is called a resolving set of $\Gamma$.It is known that the problem of computing this invariant is NP-hard. In thispaper, we consider the problem of determining the cardinality $\psi(\Gamma)$ ofminimal doubly resolving sets of $\Gamma$, and the strong metric dimension forjellyfish graph $JFG(n, m)$ and cocktail party graph $CP(k+1)$.

Metric dimension, minimal doubly resolving sets and the strong metric dimension for jellyfish graph and cocktail party graph