Graph Notations
Let (S, &Gamma) be a graph and b &isin S be a node. [Our notation generally follows that used in Marvin Schaefer, A Mathematical Theory of Global Program Optimization, Prentice-Hall, 1973.]
&Gamma b = { x &isin S | (b, x) &isin &Gamma }
are the nodes that are immediate successors of b .
&Gamma+ b = { x &isin S | (b, x) &isin &Gamma+ }
are the nodes that are successors of b .
&Gamma-1 b = { x &isin S | (x, b) &isin &Gamma }
are the nodes that are immediate predecessors of b .
Let A &sub S be a subset of the set of nodes S .
&Gamma A = { y &isin S | (x, y) &isin &Gamma
&and x &isin A }
is the set of nodes that are immediate successors of nodes in A .
&Gamma-1 A = { x &isin S | (x, y) &isin &Gamma
&and y &isin A }
is the set of nodes that are immediate predecessors of nodes in A .
We say (A, &GammaA) is a subgraph of (S, &Gamma) , where
&GammaA x = &Gamma x &cap A
is the set of transitions within the subgraph.