On double signal number of a graph

A set $S$ of vertices in a connected graph ${G=(V,E)}$ is called a signal set if every vertex not in $S$ lies on a signal path between two vertices from $S$. A set $S$ is called a double signal set of $G$ if $S$ if for each pair of vertices $x,y \in G$ there exist $u,v \in S$ such that $x,y \in L[u,v]$. The double signal number $\mathrm{dsn}\,(G)$ of $G$ is the minimum cardinality of a double signal set. Any double signal set of cardinality $\mathrm{dsn}\,(G)$ is called $\mathrm{dsn}$-set of $G$. In this paper we introduce and initiate some properties on double signal number of a graph. We have also given relation between geodetic number, signal number and double signal number for some classes of graphs.