On some extremal position problems for graphs

Tuite, James; Thomas, Elias; S.V., Ullas Chandran

doi:10.26493/1855-3974.3094.bc6

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…The following family of graphs T * (r, a) from [25] yields another family of graphs with unimodal general position polynomial. Take a complete a-partite graph, each part of which contains r vertices and label the vertices in part i by i 1 , .…”

Section: Results Mathmentioning

confidence: 99%

General Position Polynomials

Iršič,

Klavžar,

Rus

et al. 2024

Results Math

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A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as $$\sum _{i \ge 0} a_i x^i$$ ∑ i ≥ 0 a i x i , where $$a_i$$ a i is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K(n, 2), with unimodal general position polynomials are presented.

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Section: Results Mathmentioning

confidence: 99%

General Position Polynomials

Iršič,

Klavžar,

Rus

et al. 2024

Results Math

Self Cite

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“…It was shown in [39] that for any graph G we have mp(G) ≤ gp(G) and that for any 2 ≤ a ≤ b there exists a graph with mp(G) = a and gp(G) = b. This question was explored further in [41], which asked for the smallest possible order of a graph G with mp(G) = a and gp(G) = b for given a ≤ b.…”

Section: Connection With Lower Monophonic Position Numbermentioning

confidence: 99%

Lower general position sets in graphs

Di Stefano,

Klavžar,

Krishnakumar

et al. 2024

Discuss. Math. Graph Theory

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A subset S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position number gp − (G) of G, which is the number of vertices in a smallest maximal general position set of G. We show that gp − (G) = 2 if and only if G contains a universal line and determine this number for several classes 2 Di Stefano, Klavžar, Krishnakumar, Tuite and Yero of graphs, including Kneser graphs K(n, 2), line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.

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On some extremal position problems for graphs

Cited by 2 publications

References 17 publications

General Position Polynomials

General Position Polynomials

Lower general position sets in graphs

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