Investigating the Vertex Fixed Odd Geo-Domination Number in Graph Theory

Abstract

Let x be any vertex of a connected graph G of order n ≥ 3. A set S ⊆ V of a graph G is said to be an x - odd geo-dominating set if for every vertex ???? ∈ ???? − (???? ???? {????}) must lies in x – y geodesic for some y ∈ S and |????(????) ∩ (???? ???? {????})| Ξ 1(???????????? 2). The minimum cardinality of the x - odd geo-dominating set is called x - odd geo-domination number denoted by ????????−????????????(????). The x - odd geo-
dominating set with cardinality ????????−????????????(????) is called ????????−????????????– set of G. In this paper we determine the bounds for a vertex fixed odd geodomination number and the same for some families of graphs It is shown that every pair k, n of integers with 1 ≤ k ≤ n – 1,???? ≥ 3,
there exists a connected graph G of order n and ????????−????????????(????) = ????