New types of graph ideal approximation spaces.

The main aim of this paper is to combine the connections between graph theory and rough set theory. We use the notion of upper vertex set Ξ((C(ℶ))) on a subgraph ℶ of a given graph ⟲ to define the graph local function and its related graph closure operator clΞ∗ ((C(ℶ))) in the graph ideal approximation space (⟲,Ξ,L). This, graph ideal approximation space (⟲,Ξ,L) based on a graph ideal L joined to the graph approximation space (⟲,Ξ) are introduced as well. In both the graph approximation space and the graph ideal approximation spaces, the separation axioms Ti, i = 0,1,2 are reformulated. The definitions are shown using examples. Concepts of connectedness in graph ideal approximation spaces are presented together with an explanation of their distinctions. However, simple directed graphs with or without loops are the ones that are discussed throughout the study. The obtained results are valid for any type of graph: multi-graphs or simple graphs, connected or disconnected graphs, with loops or without loops, and undirected or directed graphs.