The general position number of integer lattices

Klavžar, Sandi; Rus, Gregor

doi:10.1016/j.amc.2020.125664

Cited by 14 publications

(10 citation statements)

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“…The result gp(P ∞ P ∞ ) = 4 was recently generalized in two directions. In [9], it is proved that gp(P ,k ∞ ) = 2 2 k−1 , while in [24], it is demonstrated that if T 1 and T 2 are arbitrary trees, then gp(T 1 T 2 ) = gp(T 1 ) + gp(T 2 ). The general position number of several other Cartesian product graphs is studied in [23].…”

Section: Introductionmentioning

confidence: 99%

On General Position Sets in Cartesian Products

et al. 2021

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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.

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Section: Introductionmentioning

confidence: 99%

On General Position Sets in Cartesian Products

et al. 2021

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“…In [3], general position sets in graphs were characterized. Several additional papers on the concept followed, many of them dealing with bounds on the general position number and exact results in product graphs, Kneser graphs, and more, see [11,17,18,22,24,[26][27][28]. In addition, the concept was very recently extended to the Steiner general position number [16].…”

Section: Introductionmentioning

confidence: 99%

Zero forcing number versus general position number in tree-like graphs

Hua¹,

Hua²,

Klavžar³

2021

Preprint

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Let Z(G) and gp(G) be the zero forcing number and the general position number of a graph G, respectively. Known results imply that gp(T ) ≥ Z(T )+1 holds for every nontrivial tree T . It is proved that the result extends to block graphs. For connected, unicyclic graphs G it is proved that gp(G) ≥ Z(G). The result extends neither to bicyclic graphs nor to quasi-trees. Nevertheless, a large class of quasi-trees is found for which gp(G) ≥ Z(G) holds.

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“…A couple of years earlier and in a different language, an equivalent problem was posed in [5]. A subset S of vertices in a graph G is a general position set if no triple of vertices from S lie on a common geodesic in G. The General Position problem is to find a largest general position set of G, the order of such a set is the general position number gp(G) of G. Since its introduction, the general position number has been studied for several graph classes (e.g., grid networks [19], cographs and bipartite graphs [3], graph classes with large general position number [26], Cartesian products of graphs [15,16,27,28], and Kneser graphs [13,22]).…”

Section: Introductionmentioning

confidence: 99%

On the mutual visibility in Cartesian products and triangle-free graphs

Cicerone¹,

Stefano²,

Klavžar³

2021

Preprint

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Given a graph G = (V (G), E(G)) and a set P ⊆ V (G), the following concepts have been recently introduced: (i) two elements of P are mutually visible if there is a shortest path between them without further elements of P ; (ii) P is a mutual-visibility set if its elements are pairwise mutually visible; (iii) the mutual-visibility number of G is the size of any largest mutual-visibility set. In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and investigate independent mutual-visibility sets. Then, we characterize the triangle-free graphs with the mutual-visibility number equal to 3.

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The general position number of integer lattices

Cited by 14 publications

References 14 publications

On General Position Sets in Cartesian Products

On General Position Sets in Cartesian Products

Zero forcing number versus general position number in tree-like graphs

On the mutual visibility in Cartesian products and triangle-free graphs

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