Extremal functions for shortening sets of paths

Let P-1,...,P-k be k vertex-disjoint paths in a graph G where the ends of P-i are x(i), and y(i). Let H be the subgraph induced by the vertex sets of the paths. We find edge bounds E-1(n), E-2 (n) such that: (1) if e(H) >= E-1 (vertical bar V(H)vertical bar), then there exist disjoint paths P-1',....,P-k'. where the ends of P-i' are x(i) and y(i) such that vertical bar boolean OR(i) V(P-i)vertical bar > vertical bar boolean OR(i) V(P-i')vertical bar; (2) if e(H) > E-2 (vertical bar V(H)vertical bar), then there exist disjoint paths P-l',..., P-k' where the ends of P-l' are x(i)' and y(i)' such that vertical bar boolean OR U-i V(P-i)vertical bar > vertical bar boolean OR(i) V(P-i')l and {x(1),...,x(k)} = {x(j)',...,x(k)'} and {y(l),...,y(k)) = {y(l)',...,y(k)'] The bounds are the best possible, in that there exist arbitrarily large graphs H' with e(H') = E-i(H') - 1 without the properties stipulated in I and 2.