Let us first consider the case of [x.sub.13] = 0, i.e., the

root vertex [v.sub.13] of the tree is not installed with a facility.

A directed tree is a directed graph such that each vertex has only one parent vertex, except one special vertex (

root vertex) without any parent vertex.

At each step, a

root vertex is picked among the unaggregated vertices; then, an aggregate is formed with all the unaggregated vertices that have a path of length at most r to the root, where r is a given parameter.

As a consequence of Lemma 19, note that the root edge of every D gadget in any AB-graph G remains bad, that is, no matter whether other edges are incident to the

root vertex r in G.

For generalization we introduce the following notations for a tree T with

root vertex [v.sub.s].

Referring to the process of local modularity [18, 21], our algorithm selects the

root vertex firstly and visits the adjacent nodes in next layer.

Let [T.sub.k] be an olive tree with branches [t.sub.1], [t.sub.2], ..., [t.sub.k], where k [greater than or equal to] 3 and let v be its

root vertex. Then, [l.sub.v] = k and for all u [member of] V\{v}, [l.sub.u] = 1.

A directed tree is a directed graph where every vertex except the

root vertex has exactly one parent vertex and the

root vertex can be connected to any other vertices via paths.

Let T be a labelled tree with a

root vertex [v.sub.[empty set]], labelled [empty set], the rest of the vertices labelled with the alphabet {[[x.sub.0], [x.sub.1], ...

Since each vertex (respectively, arc) in G can be attributed to one vertex (respectively, arc) in exactly one augmented graph, and there is one extra

root vertex for each augmented graph, Expression (1) -- O(n + m).

We denote a monitored entity (ME) as [T.sub.v], where [T.sub.v] contains a

root vertex v and any edge of G belongs to one and only one [T.sub.v].

In a tree with diameter D, no vertex is more than D/2 hops or edges from the

root vertex of the tree [17].