On the Power Domination Number of de Bruijn and Kautz Digraphs

Grigorious, Cyriac; Kalinowski, Thomas; Stephen, Sudeep

doi:10.1007/978-3-319-78825-8_22

Cited by 5 publications

(8 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As a consequence, our definition of power domination differs from the one in [19] in the following situation. If there is a loop on a white vertex v and all vertices in N + (v) \ {v} are blue, by our rules v becomes blue, while by the rules in [19] v remains white. By treating digraphs with loops in the same way as in zero forcing, our definition preserves an important relationship between zero forcing and power domination in digraphs without loops, also present in the case of undirected graphs (see [7]).…”

Section: If G Does Not Have Any Loops Then For Everymentioning

confidence: 94%

“…It is important to remark that in [19], the authors studied power domination in digraphs with loops using the rules introduced in [1] for digraphs without loops, while we defined power domination using different rules for digraphs with at least one loop than for digraphs without loops. As a consequence, our definition of power domination differs from the one in [19] in the following situation. If there is a loop on a white vertex v and all vertices in N + (v) \ {v} are blue, by our rules v becomes blue, while by the rules in [19] v remains white.…”

Section: If G Does Not Have Any Loops Then For Everymentioning

confidence: 99%

“…The power domination problem on digraphs was formally introduced in [1], and while not mentioned in the definition itself ([1, Definition 3.3.1]), it is clearly stated in [1,Pg.4] that only digraphs without loops were considered. Prior to our work, power domination in digraphs with loops had only been studied in [19]. It is important to remark that in [19], the authors studied power domination in digraphs with loops using the rules introduced in [1] for digraphs without loops, while we defined power domination using different rules for digraphs with at least one loop than for digraphs without loops.…”

Section: Introductionmentioning

confidence: 99%

“…Further results on zero forcing on digraphs can be found in [5], [15], [21], [23] and [31]. Power domination in digraphs has not been so thoroughly explored, but recently zero forcing and power domination for de Bruijn and Kautz digraphs was studied in [19]. De Bruijn and Kautz digraphs are iterated line digraphs of the complete digraph, with and without loops, respectively.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Zero forcing in iterated line digraphs

Ferrero

Kalinowski

Stephen

2019

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks.In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.1 in network science, it models the spread of a disease over a population, or of an opinion in a social network (see [14]).In addition to its intrinsic relation to minimum rank, zero forcing is closely related to power domination, a graph theory concept introduced in [20] to optimize the monitoring process of electrical power networks. From the definitions of power domination and zero forcing, it follows that the closed out neighborhood of a power dominating set is a zero forcing set, and a stronger relationship between zero forcing and power domination was established in [7].Zero forcing, minimum rank and power domination are all NP-hard problems, as proven in [1], [10], and [20], respectively. Thus, it is important to obtain bounds for the minimum rank, the zero forcing and the power domination numbers, as well as closed formulas to calculate them for families of graphs. Although power domination and zero forcing where introduced on undirected graphs, they were extended to digraphs in [1] and [5], respectively. Further results on zero forcing on digraphs can be found in [5], [15], [21], [23] and [31]. Power domination in digraphs has not been so thoroughly explored, but recently zero forcing and power domination for de Bruijn and Kautz digraphs was studied in [19]. De Bruijn and Kautz digraphs are iterated line digraphs of the complete digraph, with and without loops, respectively. In this work, we extend the results in [19] to zero forcing and power domination of iterated line digraphs of any regular digraph.The line digraph has been used in a broad range of disciplines, but its large number of applications precludes us from including an exhaustive summary here. In the context of this work, since the line digraph of a digraph described by a unitary matrix can also be described by a unitary matrix, iterated line digraphs are used to obtain arbitrarily large digraphs described by unitary matrices (see [24] and [28]). Such digraphs are frequently used to model quantum systems in physics, chemistry, and engineering (see [22]), and...

show abstract

Section: If G Does Not Have Any Loops Then For Everymentioning

confidence: 94%

Section: If G Does Not Have Any Loops Then For Everymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Zero forcing in iterated line digraphs

Ferrero

Kalinowski

Stephen

2019

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…We here present the other families which were considered, without much details. For De Bruijn graphs and Kautz graphs, upper bounds on the power domination number were first given by Kuo and Wu in [23], then the exact values were characterized by Grigorious, Kalinowski and Stephen in [19].…”

Section: Other Familiesmentioning

confidence: 99%

Power Domination in Graphs

Dorbec

2020

Topics in Domination in Graphs

View full text Add to dashboard Cite

In this chapter, we are interested in power domination in graphs. Power domination is a variation of domination introduced to address a physical problem of monitoring a network with phasor measurement units. The originality of this variation is that some propagation happens, and the set of covered vertices results from an iterative process. We present a survey of known results on this specific parameter.

show abstract

Distance‐layer structure of the De Bruijn and Kautz digraphs: Analysis and application to deflection routing

2023

View full text Add to dashboard Cite

In this article, we present a detailed study of the reach distance‐layer structure of the De Bruijn and Kautz digraphs, and we apply our analysis to the performance evaluation of deflection routing in De Bruijn and Kautz networks. Concerning the distance‐layer structure, we provide explicit polynomial expressions, in terms of the degree of the digraph, for the cardinalities of some relevant sets of this structure. Regarding the application to defection routing, and as a consequence of our polynomial description of the distance‐layer structure, we formulate explicit expressions, in terms of the degree of the digraph, for some probabilities of interest in the analysis of this type of routing. De Bruijn and Kautz digraphs are fundamental examples of digraphs on alphabet and iterated line digraphs. If the topology of the network under consideration corresponds to a digraph of this type, we can perform, in principle, a similar vertex layer description.

show abstract

On the Power Domination Number of de Bruijn and Kautz Digraphs

Cited by 5 publications

References 20 publications

Zero forcing in iterated line digraphs

Zero forcing in iterated line digraphs

Power Domination in Graphs

Distance‐layer structure of the De Bruijn and Kautz digraphs: Analysis and application to deflection routing

Contact Info

Product

Resources

About