on the 2-independence subdivision number of graphs

A subset s of vertices in a graph 𝐺=(𝑉,𝐸) is 2 independent if everyvertex of s has at most one neighbor in s. the 2 independence numberis the maximum cardinality of a 2 independent set of 𝐺. in this paper,we initiate the study of the 2 independence subdivision number mathrm 𝑠𝑑β2(𝐺) defined as the minimum number of edges that must besubdivided (each edge in 𝐺 can be subdivided at most once) in order toincrease the 2 independence number. we first show that for every connectedgraph 𝐺 of order at least three, 1≤ mathrm 𝑠𝑑β2(𝐺) ≤ 2, and we give a necessary and sufficient condition for graphs 𝐺 attainingeach bound. moreover, restricted to the class of trees, we provide aconstructive characterization of all trees t with mathrm 𝑠𝑑β2(𝑇)=2, and we show that such a characterization suggests an algorithmthat determines whether a tree t hastextrm mathrm 𝑠𝑑β2(𝑇)=2 or mathrm 𝑠𝑑β2(𝑇)=1 in polynomial time.