DOUBLY ISOLATED EDGE GEODETIC NUMBER OF A GRAPH

Source abstract

Here we study the new concept of a doubly isolated edge geodetic set for a connected graph 𝐺 = (𝑉, 𝐸) of order 𝑛 ≥ 3 . The edge geodetic set 𝑆 ⊆ 𝑉 of a connected graph 𝐺 is said to be a doubly isolated edge geodetic set if the induced subgraphs < 𝑆 > and < 𝑉(𝐺) − 𝑆 > have at least one isolated vertex. The doubly isolated edge geodetic number is the minimum cardinality of a doubly isolated edge geodetic set of G and is denoted by 𝑔1𝑑𝑖(𝐺). The doubly isolated edge geodetic number of some standard graphs, Cartesian product, and corona graphs is determined. It is shown that for any three positive integers 𝑎, 𝑏 and 𝑐 with 𝑎 ≤ 𝑏 ≤ 𝑐 there exists a connected graph 𝐺 of order 𝑛 with 𝑔(𝐺) = 𝑎, 𝑔1(𝐺) = 𝑏 and 𝑔1𝑑𝑖(𝐺) = 𝑐.