A　graph　G　is　diameter-2-critical　if　its　diameter　is　2,　and　the　deletion　of　any　edge　strictly　increases　the　diameter.　The　longstanding　Murty-Simon　Conjecture　states　that　for　any　diameter-2-critical　graph　G　of　order　n,　e(G)　＜=　left　perpendicular　n(2)/4　right　perpendicular　and　the　unique　extremal　graph　is　K-inverted　right　perpendicular　n/2　inverted　left　perpendicular,K-　left　perpendicular　n/2　right　perpendicular.　After　more　than　ten　years,　Furedi　(JGT,1992)　confirmed　the　Murty-Simon　Conjecture　for　sufficiently　large　n.　In　the　same　paper,　Furedi　claimed　(without　proof)　that　all　diameter-2-critical　graphs　with　at　least　left　perpendicular　(n-1)(2)/　4　right　perpendicular　+1　edges　are　complete　bipartite　graphs　and　M,　where　M　is　obtained　by　subdividing　one　edge　of　K-inverted　right　perpendicular　((n-1)/2　inverted　left　perpendicular,　left　perpendicular　(n-1)/2　right　perpendicula).　Later,　Balbuena,　Hansberg,　Haynes,　and　Henning　(Graphs　Combin.,　2015)　presented　a　class　of　diameter-2-critical　graphs　containing　M.　And　all　of　them　have　left　perpendicular　(n-1)(2)/　4　right　perpendicular　+1　edges.　So　Furedi＇s　claim　needs　to　be　revised.　In　this　paper,　we　prove　that　all　C5-free　diameter2-critical　graphs　with　at　least　left　perpendicular　(n-1)(2)/　4　right　perpendicular　+1　+　1　edges　are　complete　bipartite　graphs　for　sufficiently　large　n.　(c)　2025　Elsevier　B.V.　All　rights　are　reserved,　including　those　for　text　and　data　mining,　AI　training,　and　similar　technologies.