UDC 519.1
In 2002, Gary Chartrand and Ping Zhang [<em>The Steiner number of a graph,</em> Discrete Math., <strong>242</strong>, 41--54 (2002)] characterized the connected graphs
G
of order
p
≥
3
with Steiner number
p
,
p
-
1
,
or
2.
In our paper, we characterize all connected graphs
G
of order
p
≥
4
with Steiner number
s
(
G
)
=
p
-
2
. In addition, we obtain some sharp Nordhaus–Gaddum bounds for the Steiner number of connected graphs whose complement is also connected.