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)≤⌊[Formula　presented]⌋　and　the　unique　extremal　graph　is　K⌈n/2⌉,⌊n/2⌋.　After　more　than　ten　years,　Füredi　(JGT,1992)　confirmed　the　Murty–Simon　Conjecture　for　sufficiently　large　n.　In　the　same　paper,　Füredi　claimed　(without　proof)　that　all　diameter-2-critical　graphs　with　at　least　⌊[Formula　presented]⌋+1　edges　are　complete　bipartite　graphs　and　M,　where　M　is　obtained　by　subdividing　one　edge　of　K⌈(n−1)/2⌉,⌊(n−1)/2⌋.　Later,　Balbuena,　Hansberg,　Haynes,　and　Henning　(Graphs　Combin.,　2015)　presented　a　class　of　diameter-2-critical　graphs　containing　M.　And　all　of　them　have　⌊[Formula　presented]⌋+1　edges.　So　Füredi＇s　claim　needs　to　be　revised.　In　this　paper,　we　prove　that　all　C5-free　diameter-2-critical　graphs　with　at　least　⌊[Formula　presented]⌋+1　edges　are　complete　bipartite　graphs　for　sufficiently　large　n.　©　2025　Elsevier　B.V.