Geodetic Number of Soft Graphs of Petersen Graph

Abstract

Let ???? ∗ = (V, E) be a simple graph and A⊆????(???? ∗ ) be any non-empty set of parameters. Let ???? be an arbitrary relation from A to V where (F, A) and (K, A) are soft sets over V and E respectively. H(a) = (F(a), K(a)) is an induced subgraph of???? ∗ for all a ∈A.Then,(???? ∗ , F, K, A) = {H(a)/???? ∈ A }≅ ∪ ????∈ A ????(????) is called the soft graph of ???? ∗ corresponding to the parameter set A and the relation ????. It is said to be a T1- soft graph of ???? ∗only if H(a) is connected ∀ ???? ∈ A.Otherwise it is called a T12-soft graph of ???? ∗ . Every T1-soft graph is also a T12-soft graph of ???? ∗ and not the converse. The geodetic set of (???? ∗ , F, K, A)is introduced by K Palani et al.[7] and is defined as the union of geodetic sets of the induced sub graphs H(a) where a ∈ A. A geodetic set of a T1 or T12- soft graph of ???? ∗ of minimum cardinality is said to be a minimum geodetic set of (???? ∗ , F, K, A). The geodetic number of (???? ∗ , F, K, A) is the cardinality of a minimum geodetic set of (???? ∗ , F, K, A). The geodetic number of (???? ∗ , F, K, A) is denoted as g[(???? ∗ , F, K, A)] whereas the geodetic number of any graph G is g(G).This paper analyses the soft graphs of Petersen graphand geodeticnumber of different soft graphs of Petersen graph.