Note on upper bound graphs and forbidden subposets

In the upper bound graph of a poset PP, the vertex set is V(P)V(P) and xyxy is an edge if there exists an m∈V(P)m∈V(P) with x,y≤Pmx,y≤Pm. We show some characterizations on split upper bound graphs, threshold upper bound graphs and difference upper bound graphs in terms of mm-subposets and canonical posets.