Characterization of classes of graphs with large general position number

Thomas, Elias John; Chandran, S. V. Ullas

doi:10.1016/j.akcej.2019.08.008

Cited by 19 publications

(12 citation statements)

References 7 publications

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“…Proof. Graphs with order n and gp-number n − 2 were classified in [19], so we merely state the result for g = 4. Then G has gp(G) = n − 2 if and only if it is isomorphic to a U(r, s) or a V (r).…”

Section: The Diameters Of Graphs With Given Order and Mp-numbermentioning

confidence: 99%

On some extremal position problems for graphs

Tuite¹,

Thomas²,

V.³

2021

Preprint

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The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with 'induced path' in place of 'geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Turán problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.

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Section: The Diameters Of Graphs With Given Order and Mp-numbermentioning

confidence: 99%

On some extremal position problems for graphs

Tuite¹,

Thomas²,

V.³

2021

Preprint

Self Cite

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“…Following the seminal papers, the general position problem has been investigated in a sequence of papers [1,6,10,14,16,18,22,26]. As it happens, in the special case of hypercubes, the general position problem was studied back in 1995 by Körner [11] related to some coding theory problems.…”

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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“…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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Characterization of classes of graphs with large general position number

Cited by 19 publications

References 7 publications

On some extremal position problems for graphs

On some extremal position problems for graphs

On General Position Sets in Cartesian Products

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

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