Minimum Size of Some Metrics for the Graph $H(n)$, the Line Graph of the Graph $H(n)$ and the Cartesian product $C_n\Box P_k$

Abstract

For an arranged subset $Q = \{q_1, q_2, ..., q_k\}$ of vertices in a connected graph $G$ the metric representation of a vertex $v$ in $G$, is the $k$-vector $r(v | Q) = (d(v, q_1), d(v, q_2), ..., d(v, q_k ))$ relative to $Q$. Also, the subset $Q$ is considered as resolving set for $G$ if any pair of vertices of $G$ is distinguished by some vertices of $Q$. In the present article, we study the minimum size of resolving set, and doubly resolving set for the graph $H(n)$, and the line graph of the graph $H(n)$ is denoted by $L(n)$. Also, we compute some metrics for the Cartesian product $C_n\Box P_k$ based on the resolving sets in graphs. It is well known that these problems are NP hard