General Position Sets, Colinear Sets, and Sierpiński Product Graphs

Let \(G \otimes _f H\) $G{\otimes }_{f}H$ denote the Sierpiński product of graphs G and H with respect to the function f. The Sierpiński general position number \(\textrm{gp}{_{\textrm{S}}}(G,H)\) $\text{gp}_{\text{S}}^{}(G,H)$ is introduced as the cardinality of a largest general position set in \(G \otimes _f H\) $G{\otimes }_{f}H$ over all possible functions f. Similarly, the lower Sierpiński general position number \(\underline{\textrm{gp}}{_{\textrm{S}}}(G,H)\) $\underline{\text{gp}}_{\text{S}}^{}(G,H)$ is the corresponding smallest cardinality. The concept of vertex-colinear sets is introduced. Bounds for the general position number in terms of extremal vertex-colinear sets, and bounds for the (lower) Sierpiński general position number are proved. The extremal graphs are investigated. Formulas for the (lower) Sierpiński general position number of the Sierpiński products with \(K_2\) $K_2$ as the first factor are deduced. It is proved that if \(m,n\ge 2\) $m,n\ge 2$ , then \(\textrm{gp}{_{\textrm{S}}}(K_m,K_n) = m(n-1)\) $\text{gp}_{\text{S}}^{}(K_m,K_n)=m(n-1)$ , and that if \(n\ge 2\,m-2\) $n\ge 2m-2$ , then \(\underline{\textrm{gp}}{_{\textrm{S}}}(K_m,K_n) = m(n-m+1)\) $\underline{\text{gp}}_{\text{S}}^{}(K_m,K_n)=m(n-m+1)$ .