Projects in (t, r) Broadcast Domination

Harris, Pamela E.; Insko, Erik; Johnson, Katie

doi:10.1007/978-3-030-37853-0_8

Cited by 2 publications

(4 citation statements)

References 10 publications

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“…This parameter has been studied on small grid graphs in [10], and its density has been studied on infinite grid graphs in [7] [9] [11], on the King's lattice in [12], and on triangular finite and infinite graphs in [13]. Moreover, [14] features many open problems for future study in this area.…”

Section: Introductionmentioning

confidence: 99%

On (<i>t, r</i>) Broadcast Domination of Directed Graphs

Harris¹,

Hollander²,

Insko³

2022

OJDM

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A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases 1 r = and ( ) ( ) , 2, 2 t r = , for every integer value in this interval, there exists an orientation G  of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.

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Section: Introductionmentioning

confidence: 99%

On (<i>t, r</i>) Broadcast Domination of Directed Graphs

Harris¹,

Hollander²,

Insko³

2022

OJDM

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“…This parameter has been studied on small grid graphs in [15], and its density has been studied on infinite grid graphs in [3,6], on the King's lattice in [5], and on triangular finite and infinite graphs in [9]. Moreover, [8] features many open problems for future study in this area.…”

Section: Introductionmentioning

confidence: 99%

“…Figure 4. For (t, r) =(10,8), the strategy used in Lemma 3.1 does not hold. In the following example, the resulting set of vertices of the bottom graph does not form a directed(10,8) dominating set.…”

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On $(t,r)$ broadcast domination of directed graphs

Harris,

Hollander,

Insko

2021

Preprint

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A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation G of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.

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Projects in (t, r) Broadcast Domination

Cited by 2 publications

References 10 publications

On (<i>t, r</i>) Broadcast Domination of Directed Graphs

On (<i>t, r</i>) Broadcast Domination of Directed Graphs

On $(t,r)$ broadcast domination of directed graphs

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Projects in (t, r) Broadcast Domination

Cited by 2 publications

References 10 publications

On (&lt;i&gt;t, r&lt;/i&gt;) Broadcast Domination of Directed Graphs

On (&lt;i&gt;t, r&lt;/i&gt;) Broadcast Domination of Directed Graphs

On $(t,r)$ broadcast domination of directed graphs

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On (<i>t, r</i>) Broadcast Domination of Directed Graphs

On (<i>t, r</i>) Broadcast Domination of Directed Graphs