Authors’ summary: We consider upper bound graphs and double bound graphs of posets. We obtain a characterization of upper bound graphs whose complements are also upper bound graphs as follows: for a connected graph \(G\), both \(G\) and \(\overline G\) are upper bound graphs if and only if \(G\) is a split graph with \(V(G)= K+ S\), where \(K\) is a clique and \(S\) is an independent set, satisfying one of the following conditions: (1) there exists a vertex in \(K\) with no neighbour in \(S\), or (2) for each edge \(e= uv\) in \(K\), there exists a vertex \(w\in S\) such that \(u,v\in N(w)\), and for each pair of vertices \(x,y\in S\), there exists a vertex \(v\in K\) such that neither \(x\) nor \(y\) is adjacent to \(v\). We also obtain some properties of double bound graphs of height one posets.