Let C be a longest cycle in a 3-connected graph G and let H be a component of G-C such that |H|≥3. In

subject to some condition on H

it is shown that |C|≥ 2d(u)+2d(v)-5 with strict inequality unless G belongs to some exceptional class of graphs

where u

v are non-adjacent vertices. In this thesis

we supply an explicit characterization of the exceptional class of graphs for the above estimate of |C|.