On the Outer-Independent Roman Domination in Graphs

Martínez, Abel Cabrera; García, Suitberto Cabrera; Carrión García, Andrés; Grisales del Rio, Angela María

doi:10.3390/sym12111846

Open AccessArticle

On the Outer-Independent Roman Domination in Graphs

by

Abel Cabrera Martínez

^1,*

,

Suitberto Cabrera García

²,

Andrés Carrión García

³

and

Angela María Grisales del Rio

³

¹

Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain

²

Departamento de Estadística e Investigación Operativa Aplicadas y Calidad, Universitat Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

³

Centro de Gestión de la Calidad y del Cambio, Universitat Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

^*

Author to whom correspondence should be addressed.

Symmetry 2020, 12(11), 1846; https://doi.org/10.3390/sym12111846

Submission received: 21 October 2020 / Revised: 3 November 2020 / Accepted: 6 November 2020 / Published: 9 November 2020

(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)

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Abstract

Let G be a graph with no isolated vertex and

f : V (G) \to {0, 1, 2}

a function. Let

V_{i} = {v \in V (G) : f (v) = i}

for every

i \in {0, 1, 2}

. The function f is an outer-independent Roman dominating function on G if

V_{0}

is an independent set and every vertex in

V_{0}

is adjacent to at least one vertex in

V_{2}

. The minimum weight

ω (f) = \sum_{v \in V (G)} f (v)

among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs.

Keywords: outer-independent Roman domination; Roman domination; vertex cover; rooted product graph

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MDPI and ACS Style

Martínez, A.C.; García, S.C.; Carrión García, A.; Grisales del Rio, A.M. On the Outer-Independent Roman Domination in Graphs. Symmetry 2020, 12, 1846. https://doi.org/10.3390/sym12111846

AMA Style

Martínez AC, García SC, Carrión García A, Grisales del Rio AM. On the Outer-Independent Roman Domination in Graphs. Symmetry. 2020; 12(11):1846. https://doi.org/10.3390/sym12111846

Chicago/Turabian Style

Martínez, Abel Cabrera, Suitberto Cabrera García, Andrés Carrión García, and Angela María Grisales del Rio. 2020. "On the Outer-Independent Roman Domination in Graphs" Symmetry 12, no. 11: 1846. https://doi.org/10.3390/sym12111846

APA Style

Martínez, A. C., García, S. C., Carrión García, A., & Grisales del Rio, A. M. (2020). On the Outer-Independent Roman Domination in Graphs. Symmetry, 12(11), 1846. https://doi.org/10.3390/sym12111846

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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