Graph Notations

Let (S, Γ) be a graph and b ∈ S be a node.

Γ b = { x ∈ S | (b, x) ∈ Γ }
are the nodes that are immediate successors of b .

Γ⁺ b = { x ∈ S | (b, x) ∈ Γ⁺ }
are the nodes that are successors of b .

Γ^-1 b = { x ∈ S | (x, b) ∈ Γ }
are the nodes that are immediate predecessors of b .

Let A ⊂ S be a subset of the set of nodes S .

Γ A = { y ∈ S | (x, y) ∈ Γ ∧ x ∈ A }
is the set of nodes that are immediate successors of nodes in A .

Γ^-1 A = { x ∈ S | (x, y) ∈ Γ ∧ y ∈ A }
is the set of nodes that are immediate predecessors of nodes in A .

We say (A, Γ_A) is a subgraph of (S, Γ) , where
Γ_A x = Γ x ∩ A
is the set of transitions within the subgraph.