Let \ (S\) be an independent set of a connected graph \ (G\) of order atleast \ (2\). A set \ (S V (G) -S\) is an \ (S\) -fixed geodetic set of \ (G\) if each vertex \ (v\) in \ (G\) lies on an \ (x-y\) geodesic for some \ (x S\) and \ (y S\). The \ (S\) -fixed geodetic number \ (gₛ (G) \) of \ (G\) is the minimum cardinality of an \ (S\) -fixed geodetic set of \ (G\). The independent fixed geodetic number of \ (G\) is \ (g₈₅ (G) = min \gₛ (G) \\), where the minimum is taken over all independent sets \ (S\) in \ (G\). An independent fixed geodetic set of cardinality \ (g₈₅ (G) \) is called a \ (g₈₅\) -set of \ (G\). We determine bounds for it and characterize graphs which realize these bounds. Also, the relations with the vertex geodomination number, vertex independence number and vertex covering number of graphs are studied. Some realization results based on the parameter \ (g₈₅ (G) \) are generated. Finally, two algorithms are designed to compute the independent fixed geodetic number \ (g₈₅ (G) \) and their complexity results are analyzed.
Titus et al. (Sun,) studied this question.