On General Position Sets in Cartesian Products

Abstract

The general position number $$\mathrm{gp}(G)$$ gp ( G ) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G. The general position number of cylinders $$P_r\,\square \,C_s$$ P r □ C s is deduced. It is proved that $$\mathrm{gp}(C_r\,\square \,C_s)\in \{6,7\}$$ gp ( C r □ C s ) ∈ { 6 , 7 } whenever $$r\ge s \ge 3$$ r ≥ s ≥ 3 , $$s\ne 4$$ s ≠ 4 , and $$r\ge 6$$ r ≥ 6 . A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $$P_r\,\square \,P_s$$ P r □ P s , where $$r,s\ge 2$$ r , s ≥ 2 , is also determined.

Extremal edge general position sets in some graphs

“…Edge general position sets in graphs occur as the edge variant of general position sets. The latter sets have already been extensively researched, see the seminal papers [6,12], some of the subsequent ones [3,9,10,11,13], and references therein. The edge version has been introduced in [7] and further studied in [8].…”

Edge general position problem

“…Following the above listed seminal papers, the general position problem has been extensively investigated [1,9,13,15,16,20,25,27,29,30,31]. Let us emphasize some selected results.…”

“…In the same paper it was also proved that 10 ≤ gp(P 3 ∞ ) ≤ 16. The lower bound 10 was improved to 14 in [13]. These efforts culminated in [15] where it is proved that if n ∈ N, then gp(P n ∞ ) = 2 2 n−1 .…”

“…These efforts culminated in [15] where it is proved that if n ∈ N, then gp(P n ∞ ) = 2 2 n−1 . The general position problem in Cartesian products has been further investigated in [13,30,31]. In particular, it was proved in [30] that gp(G H) ≤ n(G) + n(H) − 2, where the equality holds if and only if G and H are both generalized complete graphs.…”

On the general position set of two classes of graphs

“…In [11], Manuel and Klavžar determined some upper bounds on gp(G) and showed that the general position problem is NP-complete. For more properties of the general position problem, please see [9,13]. Since researchers are very concerned about the cactus and wheel graphs, see [6,18,16] for examples, it is interesting to study the bounds on the gp-number for cactus and wheel graphs.…”