** 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

* &Gamma ^{-1} b = { x &isin S | (x, b) &isin &Gamma } *

are the nodes that are

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

We say * (A, &Gamma _{A}) * is a

is the set of transitions within the subgraph.