Note on strict-double-bound numbers of paths, cycles, and wheels (Q2839745)

The paper analyses the graph invariant \textit{strict-double-bound number} \(\zeta(G)\) defined as \(\min\{n; G\cup N_n\) is a strict-double-bound graph\(\}\), where \(N_n\) is the empty graph on \(n\) vertices and for a poset \(P=(X,\leq_p)\) we define \textit{strict-double-bound graph} of \(P\) as the graph \(sDB(P)\) on \(X\) for which vertices \(u\) and \(v\) are adjacent if and only if \(u\neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) s.t. \(x\leq u\leq y\) and \(x\leq v\leq y\).NEWLINENEWLINEThis paper focuses on some special classes of graphs, namely paths and wheels, and obtains their \textit{strict-double-bound number}. For sums \(G+K_n\) for any graph \(G\) without isolated vertices it is proven that \(\zeta(G+K_n) = \zeta(G)\).