Note on strict-double-bound numbers of nearly complete graphs missing some edges

Abstract

For a poset $P=(X,{\le }_P)$, the strict-double-bound graph (strict DB-graph $\mathrm{sDB}\left(P\right)$) is the graph on $X$ for which $u$ is adjacent to $v$ if and only if $u\ne v$ and there exist elements $x,y\in X$ distinct from $u$ and $v$ such that $x\le u\le y$ and $x\le v\le y$. The strict-double-bound number $\zeta \left(G\right)$ of a graph $G$ is defined as $min\{l;\mathrm{sDB}\left(P\right)\cong G\cup \overline{K_l}\text{ for some poset }P\}$.

We obtain strict-double-bound numbers of nearly complete graphs missing one, two or three edges. In particular, we prove that $\zeta (K_n-e)=3,\zeta (K_n-E\left(P_3\right))=3,\zeta (K_n-E\left(2K_2\right))=4,\zeta (K_n-E\left(K_3\right))=4,\zeta (K_n-E\left(P_4\right))=4,\zeta (K_n-E\left(K_{1,3}\right))=3,\zeta (K_n-E(P_3\cup K_2))=4$ and $\zeta (K_3-E\left(3K_2\right))=5$.