We now consider the case where all the links affected by a risk share an end-vertex.

putting it in the middle of the subpath replacing the edge or attaching it at the other end-vertex.

If v [member of] V(G) and e [member of] E(G), we use v ~ e to denote that v is an

end-vertex of e.

Furtula, Topological index based on the ratios of geometrical and arithmetical means of

end-vertex degrees of edges, J.

If max{d(x),d(y)}[greater than or equal to]c/2 for any two vertices u and v with d(u,v)=2, then G has a longest path with at least an end-vertex x satisfying d(x)[greater than or equal to]c/2 or G is a Hamiltonian.

1])[greater than or equal to]c/2, then Pm is the longest path of G of order m with its end-vertex [x.

The removal of an

end-vertex of the edges linking two cliques (e.

Thus, an edge of Fab has one

end-vertex with zero at the first position and one

end-vertex with one at the first position.

Now, assume that deg(v) [greater than or equal to] 3 that means v is end-vertex of a maximal simple path.

We have no new adjacent vertices with the same color, unless the other end-vertex of [p.

Since any generic search traverses the connected components of a graph one by one in any order, if we want to know whether or not a vertex t can be the last vertex of some search S on some graph G, it is enough to consider the

end-vertex problem on the graph induced by the connected component of G that contains t.

All the edges with one

end-vertex equal to a belong to different [THETA] classes.