On Strict-Double-Bound Numbers of Spiders and Ladders

Abstract

For a poset \(P = (X, \leq_ P)\), the strict-double-bound graph (\(sDB\)-graph \(sDB(P)\)) is the graph on \(X\) for which vertices \(u\) and \(v\) of \(sDB(P)\) are adjacent if and only if \(u \neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) such that \(x \leq_ P y\) and \(x \leq_P v \leq_P y\). The strict-double-bound number \(\zeta(G)\) of a graph \(G\) is defined as \(\min\{n; G \cup \overline{K}_n \text{ is a strict-double-bound graph}\}\).

We obtain that for a spider \(S_{n,m}\) (\(n,m > 3\)) and a ladder \(L_n\) (\(n \geq 4\)), \(\left\lceil2\sqrt{nm}\right\rceil \leq \zeta(S_{n,m}) \leq n+m\), \(\zeta(S_{n,n}) = 2n\), and \(\left\lceil 2\sqrt{3n+2}\right\rceil \leq \zeta(L_n) \leq 2n\).