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Open AccessArticle

An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number

by Enqiang Zhu

Mathematics 2021, 9(4), 402; https://doi.org/10.3390/math9040402 - 18 Feb 2021

Cited by 1 | Viewed by 2304

For a graph G, its k-rainbow independent domination number, written as

γ_{rik} (G)

, is defined as the cardinality of a minimum set consisting of k vertex-disjoint independent sets [...] Read more.

For a graph G, its k-rainbow independent domination number, written as

γ_{rik} (G)

, is defined as the cardinality of a minimum set consisting of k vertex-disjoint independent sets

V_{1}, V_{2}, \dots, V_{k}

such that every vertex in

V_{0} = V (G) (\cup_{i = 1}^{k} V_{i})

has a neighbor in

V_{i}

for all

i \in {1, 2, \dots, k}

. This domination invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics and Computation 333(15), 2018: 353–361), which aims to compute the independent domination number of

G □ K_{k}

(the generalized prism) via studying the problem of integer labeling on G. They proved a Nordhaus–Gaddum-type theorem:

5 \leq γ_{ri 2} (G) + γ_{ri 2} (\bar{G}) \leq n + 3

for any n-order graph G with

n \geq 3

, in which

\bar{G}

denotes the complement of G. This work improves their result and shows that if

G ≇ C_{5}

, then

5 \leq γ_{ri 2} (G) + γ_{ri 2} (\bar{G}) \leq n + 2

. Full article

(This article belongs to the Special Issue Graphs, Metrics and Models)

13 pages, 301 KB

Open AccessArticle

Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)

by Boštjan Gabrovšek, Aljoša Peperko and Janez Žerovnik

Mathematics 2020, 8(6), 996; https://doi.org/10.3390/math8060996 - 18 Jun 2020

Cited by 14 | Viewed by 2713

Abstract

We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs

P (n, k)

for certain values of

n, k \in N

. By suitably adjusting and applying a well established technique of tropical algebra [...] Read more.

We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs

P (n, k)

for certain values of

n, k \in N

. By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs

P (n, 2)

and

P (n, 3)

thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs

P (n, 2)

. The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori. Full article

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