Let [??] [subset or equal to] T be the set of all

strongly connected components terminal in T and F' = {[xi] : Inf(A; [xi]) [member of] [??]}.

The other kind of graph important for visibility based planners is the

strongly connected component graph of the roadmap or its subgraphs.

It is a

strongly connected component where all edges are of same type.

A b-balanced cut in a graph G = (V, E) is a cut that partitions the graph into connected components (

strongly connected components in the directed case), each of which contains at most (1 - b) [multiplied by] n vertices, where n = |V|.

Consider any

strongly connected component. Consider the vertex u of the component that is visited first during depth-first search.

Since s = [delta]([s.sub.0],w) [member of] S' and S" is a terminal

strongly connected component, Inf(A; [xi]) [??] [S.sub.+].sub.

In general, one can consider

strongly connected components of T(A).

In the first level, the structural properties of a graph are studied and

strongly connected components are fused to reduce the original graph to a DAG.

The above procedure requires

strongly connected component arc capacities to be at least 2[Delta] to route the total of [Delta] flow through.

It is obtained by partitioning [??] into its

strongly connected components, and then taking the quotient digraph, (so two parts are connected by an arc if and only if there is an arc between the corresponding vertex sets in [??]).

Otherwise, use the polynomial-time algorithm from [25, Corollary 2.25] to find a small vertex subset S [subset or equal to] W in G[W] with the property that every

strongly connected component of G[W] - S has at most 3/4 [absolute value of W] vertices.

Its vertices are the

strongly connected components [C.sub.1], ...