Add an

isolated vertex [v.sub.n+1] to [G.sub.2] to get a realization of F'.

Further suppose that [[upsilon].sub.i] is a free vertex of (G, m), and G' is the induced sub-graph on the vertex set [V.sub.G] {[[upsilon].sub.i]}, along with the

isolated vertex [[upsilon].sub.i].

If x is an

isolated vertex, the (OSCL) is obviously satisfied.

If b is incident with each of [F.sub.0], then X[Q.sup.k.sub.n] - F has two components, one of which is an

isolated vertex. If b is not incident with each of [F.sub.0], then X[Q.sup.k.sub.n] - F is connected; a contradiction to that F is an edge cut of X[Q.sup.k.sub.n].

Due to the existence of the

isolated vertex, there is always at least one path of infinity length in the password graph, even if we remove the vertices with 0 degrees and calculate again; the same reason for connectivity: due to the existence of isolated clusters, the diameter for filtered password graph is still +[infinity], and the result is also listed in Table 2.

Also, if G has an

isolated vertex, [M.sub.k] (G) does not exists.

2.6 Definition [15] A vertex which is not incident with any edge is called an

isolated vertex. In other words a vertex with degree zero is called an

isolated vertex.

Let {[v.sub.1], [v.sub.2], [v.sub.3]} be the path in < S > and [v.sub.4] be a

isolated vertex in < S > .

A vertex [v.sub.j] [member of] V of interval valued neutrosophic graph G = (A, B) is said to be an

isolated vertex if there is no effective edge incident at [v.sub.j].

Let [C.sub.6] be a cycle of length six and [GAMMA] be a graph obtained by connecting an

isolated vertex to one of the vertices of [C.sub.6].

Vertex of degree zero is called an

isolated vertex. Vertex of degree one is called a pendant.

If there exist one or more groups of connected vertices from [N.sub.i0], and there is an edge between i and some

isolated vertex from [N.sub.i0], we delete the edge between i and the vertex and add an edge between i and a vertex chosen from the largestgroup randomly.