Graphs with large Steiner number

UDC 519.1 In 2002, Gary Chartrand and Ping Zhang [The Steiner number of a graph, Discrete Math., 242, 41--54 (2002)] characterized the connected graphsGof orderp≥3with Steiner numberp,p-1,or2. In our paper, we characterize all connected graphsGof orderp≥4with 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.