On strict-double-bound numbers of graphs and cut sets

For a poset P = (X, <= P), the strict-double-bound graph of P is the graph sDB(P) on V(sDB(P)) = X for which vertices u and v of sDB(P) are adjacent if and only if u not equal v and there exist elements x, y is an element of X distinct from u and v such that x <=(P) u <=(P) y and x <=(P) v <=(P) y. The strict-doublebound number zeta(G) of a graph G is defined as min{ n; sDB(P) congruent to G U (K) over bar (n) for some poset P}. We obtain an upper bound of strict-double-bound numbers of graphs with a cut-set generating a complete subgraph. We also estimate upper bounds of strict-double-bound numbers of chordal graphs. Keywords: strict-double-bound graph, strict-double-bound number, cut-set, chordal graph