Note on Strict-Double-Bound Numbers of Paths, Cycles, and Wheels

For a poset \( P = (X, \leq_P) \), the strict-double-bound graph (\(sDB\)-graph) of \( P = (X, \leq_P) \) is the graph \( sDB(P) \) 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 \( x \leq u \leq y \) and \( x \leq v \leq y \). The strict-double-bound number \( \zeta(G) \) is defined as

\[
\zeta(G) = \min \{ n \mid G \cup N_n \text{ is a strict-double-bound graph} \},
\]

where \( N_n \) is the graph with \( n \) vertices and no edges.

In this paper we deal with strict-double-bound numbers of some graphs. For example, we obtain that

\[
\zeta(P_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 2 \text{)},
\]

\[
\zeta(C_n) = \lceil 2\sqrt{n} \rceil \text{ (} n \geq 4 \text{)},
\]

\[
\zeta(W_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 5 \text{)},
\]

and

\[
\zeta(G + K_n) = \zeta(G)
\]

for a graph \( G \) with no isolated vertices.