On the $P_3$-Hull Number of Kneser Graphs

Grippo, Luciano N.; Pastine, Adrián; Torres, Pablo Daniel; Valencia-Pabon, Mario; Vera, Juan C.

doi:10.37236/9903

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…A minimum r-percolating set in G is an r-percolating set S of G satisfying m(G, r) = |S|. Bootstrap percolation is very well studied in graphs, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

3-Neighbor bootstrap percolation on grids

Hedžet,

Henning

2023

Discuss. Math. Graph Theory

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Given a graph G and assuming that some vertices of G are infected, the r-neighbor bootstrap percolation rule makes an uninfected vertex v infected if v has at least r infected neighbors. The r-percolation number, m(G, r), of G is the minimum cardinality of a set of initially infected vertices in G such that after continuously performing the r-neighbor bootstrap percolation rule each vertex of G eventually becomes infected. In this paper, we consider the 3-bootstrap percolation number of grids with fixed widths. If G is the Cartesian product P 3 P m of two paths of orders 3 and m, we prove that m(G, 3) = 3 2 (m + 1) − 1, when m is odd, and m(G, 3) = 3 2 m + 1, when m is even. Moreover, if G is the Cartesian product P 5 P m , we prove that m(G, 3) = 2m + 2, when m is odd, and m(G, 3) = 2m + 3, when m is even. If G is the Cartesian product P 4 P m , we prove that m(G, 3) takes on one of two possible values, namely m(G, 3) = 5(m+1) 3

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Section: Introductionmentioning

confidence: 99%

3-Neighbor bootstrap percolation on grids

Hedžet,

Henning

2023

Discuss. Math. Graph Theory

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“…Kneser graphs have a rich combinatorial structure [10,13], and there are many studies on this class involving colorings, independent sets, and products of graphs (see [2,5,12]). In addition, the P 3 -hull number of a Kneser graph has been recently investigated in [11], where the authors determine the exact value of the P 3 -hull number number of K(2n + k, n) for k > 1, and provide lower and upper bounds for k = 1. To the best of the authors' knowledge, however, no studies on geodetic convexity parameters are known for Kneser graphs.…”

Section: Introductionmentioning

confidence: 99%

Geodetic convexity and Kneser graphs

Bedo¹,

Leite²,

Oliveira³

et al. 2022

Preprint

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The Kneser graph K(2n+k, n), for positive integers n and k, is the graph G = (V, E) such that V = {S ⊆ {1, . . . , 2n + k} : |S| = n} and there is an edge uv ∈ E whenever u ∩ v = ∅. Kneser graphs have a nice combinatorial structure, and many parameters have been determined for them, such as the diameter, the chromatic number, the independence number, and, recently, the hull number (in the context of P 3 -convexity). However, the determination of geodetic convexity parameters in Kneser graphs still remained open. In this work, we investigate both the geodetic number and the geodetic hull number of Kneser graphs. We give upper bounds and determine the exact value of these parameters for Kneser graphs of diameter two (which form a nontrivial subfamily). We prove that the geodetic hull number of a Kneser graph of diameter two is two, except for K(5, 2), K(6, 2), and K(8, 2), which have geodetic hull number three. We also contribute to the knowledge on Kneser graphs by presenting a characterization of endpoints of diametral paths in K(2n + k, n), used as a tool for obtaining some of the main results in this work.

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“…Polynomial-time algorithms to compute the hull number of some special classes of graphs were given [1,7]. Recently, Grippo et al [10] studied the P 3 -hull number of Kneser graphs K(n, k) for n 2k + 2, whose vertices are the k-subsets of a n-set X, and two vertices v 1 and v 2 are adjacent if |v 1 ∩ v 2 | = 0. Bresȃ et al [4] dealt with the problem of computing the P 3 -hull number of Hamming graphs.…”

Section: Introductionmentioning

confidence: 99%

On the $ P_3 $-hull numbers of $ q $-Kneser graphs and Grassmann graphs

Liao¹,

Cao²,

Lu³

2022

Preprint

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Let S be an n-dimensional vector space over the finite field F q , where q is necessarily a prime power. Denote K q (n, k) (resp. J q (n, k)) to be the q-Kneser graph (resp. Grassmann graph) for k ≥ 1 whose vertices are the k-dimensional subspaces of S and two vertices v 1 and v 2 are adjacent if dim(v 1 ∩ v 2 ) = 0 (resp. dim(v 1 ∩ v 2 ) = k − 1). We consider the infection spreading in the q-Kneser graphs and the Grassmann graphs: a vertex gets infected if it has at least two infected neighbors. In this paper, we compute the P 3 -hull numbers of K q (n, k) and J q (n, k) respectively, which is the minimum size of a vertex set that eventually infects the whole graph.

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On the $P_3$-Hull Number of Kneser Graphs

Cited by 5 publications

References 21 publications

3-Neighbor bootstrap percolation on grids

3-Neighbor bootstrap percolation on grids

Geodetic convexity and Kneser graphs

On the $ P_3 $-hull numbers of $ q $-Kneser graphs and Grassmann graphs

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