On upper bound graphs whose complements are also upper bound graphs

In this paper, 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 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 wES such that u, vEN(w), and for each pair of vertices x, yES, there exists a vertex vEK such that neither x nor y is adjacent to v. We also obtain some properties of double bound graphs of height one posets.