On strict-double-bound graphs and Cartesian products of paths and cycles

For a poset P = (X, <= P), the strict-double-bound graph (sDB(P)) of P = (X,<= P) is the graph with the vertex set X such that uv is an element of E(sDB(P)) if and only if u not equal v and there exist x is an element of X and y is an element of X distinct from u and v such that x <= P u <= P y and x <= P v <= P y. For a connected graph G, the strict-double-bound number zeta(G) is defined as min{l : there exists a poset P such that sDB(P) congruent to G boolean OR N-l}, where N-l is the graph with l vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that zeta(P-n xP(m)) = 2n + 2m - 4 for n, m >= 2, zeta(C-4n x C-4m) <= 8nm + 4 for n, m >= 1, and zeta(C-n x C-m) <= 4n + 4m- 4 for n, m >= 3.