A graph

is called

-dominated

if it satisfies

for every pair

of vertices at distance 2, where

,
![$N[u]\subseteq N[x]\cup N[y]\}$](img36.png)
and

if
![$v\in N(u)\backslash (N[x]\cup N[y])$](img38.png)
, then

for

at distance

}. For a noncomplete graph

, the number

is defined as

:

and

, for a complete graph

, set

. In this paper, we prove that a 2-connected

-dominated graph

of order

is hamiltonian if

and

, moreover it is best possible.