A survey and classification of Sierpiński-type graphs

Hinz, Andreas M.; Klavžar, Sandi; Zemljič, Sara Sabrina

doi:10.1016/j.dam.2016.09.024

Cited by 61 publications

(33 citation statements)

References 75 publications

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“…Grundy domination number of For another application consider the Sierpiński graphs introduced in [23] and extensively studied by now, cf. the recent survey [20]. They are defined as follows.…”

Section: Corollary 23 If G Is a Graph Without Isolated Vertices Thenmentioning

confidence: 99%

Grundy dominating sequences and zero forcing sets

Brešar

Bujtás

Gologranc

et al. 2017

Discrete Optimization

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In a graph G a sequence v1, v2, . . . , vm of vertices is Grundy dominating if for allThe length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to N (vi) ⊆ ∪ i−1 j=1 N [vj ] or N [vi] ⊆ ∪ i−1 j=1 N (vj ). In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities.

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Section: Corollary 23 If G Is a Graph Without Isolated Vertices Thenmentioning

confidence: 99%

Grundy dominating sequences and zero forcing sets

Brešar

Bujtás

Gologranc

et al. 2017

Discrete Optimization

Self Cite

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“…Except for the intrinsic theoretical interest [65], [66], Sierpiński graphs play an important role in topology, mathematics, and computer science. For example, adding an "open edge" to each of the k extreme vertices in Sierpiński graphs yields the WK-recursive networks.…”

Section: B Sierpiński Graphsmentioning

confidence: 99%

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Zhang

et al. 2019

IEEE Trans. Cybern.

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The hierarchical graphs and Sierpiński graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are nonfractal and small-world, while the Sierpiński graphs are fractal and ''large-world.'' Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.

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“…Sierpiński triangle graphs can be defined in various ways, which originate in the Sierpiński triangle fractal [20]. Here we use the connection with Sierpiński graphs.…”

Section: Sierpiński Triangle Graphsmentioning

confidence: 99%

“…see [22] where different types of vertex and edge colorings were considered for this class of graphs (cf. also [20] for a survey on Sierpiński-type graphs in which one can also find an explanation of different terminology issues). Interestingly, the graphs in this class are subgraphs of the infinite triangular lattice, hence in the limit they yield a subgraph of the infinite triangular lattice.…”

Section: Introductionmentioning

confidence: 99%

Packing coloring of Sierpiński-type graphs

Brešar

Ferme

2018

Aequat. Math.

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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets V i , i ∈ {1, . . . , k}, where each V i is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs ST n 3 is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs S n G with respect to all connected graphs G of order 4.

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A survey and classification of Sierpiński-type graphs

Cited by 61 publications

References 75 publications

Grundy dominating sequences and zero forcing sets

Grundy dominating sequences and zero forcing sets

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Packing coloring of Sierpiński-type graphs

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