A graph is called -dominated if it satisfies
for every pair of vertices at distance 2, where
,
and
if
, 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.

Key Words: -dominated graph, quasi claw-free graph, neighborhood union, hamiltonicity.

2010 Mathematics Subject Classification: Primary 05C45.