On double bound graphs and forbidden subposets

For a poset P=(X,≤P)P=(X,≤P), the double bound graph   (DB-graph) of PP is the graph DB(P)=(X,EDB(P))DB(P)=(X,EDB(P)), where xy∈EDB(P)xy∈EDB(P) if and only if x≠yx≠y and there exist n,m∈Xn,m∈X such that n≤Px,y≤Pmn≤Px,y≤Pm. We obtain that for a subposet QQ of a poset P,QP,Q is an (nn, mm)-subposet of PP if and only if DB(Q)DB(Q) is an induced subgraph DB(P)DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (nn, mm)-subposets and double canonical posets.