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For a connected graph \( G \) of order \( n \geq 2 \), a set \( S \) of vertices of \( G \) is a geodetic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x \)-\( y \) geodesic for some elements \( x \) and \( y \) in \( S \). The geodetic number \( g(G) \) of \( G \) is the minimum cardinality of a geodetic set of \( G \). A geodetic set of cardinality \( g(G) \) is called a \( g \)-set of \( G \).
A set \( S \) of vertices of a connected graph \( G \) is an open geodetic set of \( G \) if for each vertex \( v \) in \( G \), either \( v \) is an extreme vertex of \( G \) and \( v \in S \); or \( v \) is an internal vertex of an \( x \)-\( y \) geodesic for some \( x, y \in S \). An open geodetic set of minimum cardinality is a minimum open geodetic set, and this cardinality is the open geodetic number, \( og(G) \).
A connected open geodetic set of \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) is connected. The minimum cardinality of a connected open geodetic set of \( G \) is the connected open geodetic number of \( G \) and is denoted by \( og_c(G) \).
A total open geodetic set of a graph \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) contains no isolated vertices. The minimum cardinality of a total open geodetic set of \( G \) is the total open geodetic number of \( G \) and is denoted by \( og_t(G) \). A total open geodetic set of cardinality \( og_t(G) \) is called an \( og_t \)-set of \( G \).
Certain general properties satisfied by total open geodetic sets are discussed. Graphs with total open geodetic number \( 2 \) are characterized. The total open geodetic numbers of certain standard graphs are determined. It is proved that for positive integers \( r \), \( d \), and \( k \geq 4 \) with \( r \leq d \leq 2r \), there exists a connected graph of radius \( r \), diameter \( d \), and total open geodetic number \( k \). It is also proved that for the positive integers \( a \), \( b \), and \( n \) with \( 4 \leq a \leq b \leq n \), there exists a connected graph \( G \) of order \( n \) such that \( og_t(G) = a \) and \( og_c(G) = b \).