On double bound graphs and forbidden subposets

For a poset P = ( X , ≤ P ) , the double bound graph (DB-graph) of P is the graph D B ( P ) = ( X , E D B ( P ) ) , where x y ∈ E D B ( P ) if and only if x ≠ y and there exist n , m ∈ X such that n ≤ P x , y ≤ P m . We obtain that for a subposet Q of a poset P , Q is an ( n , m )-subposet of P if and only if D B ( Q ) is an induced subgraph D B ( 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 ( n , m )-subposets and double canonical posets.