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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. |
