Italian domination in trees

Henning, Michael A.; Klostermeyer, William F.

doi:10.1016/j.dam.2016.09.035

Cited by 80 publications

(38 citation statements)

References 6 publications

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“…Similar constructive characterizations of trees for different domination related parameters or properties have been presented in e.g. Alvarado et al (2016), Henning and Klostermeyer (2017), Henning and Marcon (2016), and Rad (2017). Our characterization is based on four simple operations.…”

Section: Corollary 6 Let G = ((A B) E G ) Be a Connected Bipartite supporting

confidence: 53%

Graphs with equal domination and covering numbers

et al. 2019

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A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination number γ (G) of G is the minimum cardinality of a dominating set of G. A set C ⊆ V G is a covering set of G if every edge of G has at least one vertex in C. The covering number β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which γ (G) = β(G) is denoted by C γ =β , whereas B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to C γ =β and B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859-1867, 2013) allows to recognize all the graphs belonging to the set C γ =β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

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Section: Corollary 6 Let G = ((A B) E G ) Be a Connected Bipartite supporting

confidence: 53%

Graphs with equal domination and covering numbers

et al. 2019

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“…The sum w f = ∑ v∈V(G) f (v) is denoted the weight of a Roman {2}-dominating function, and the minimum weight of a Roman {2}-dominating function f is the Roman {2}-domination number, denoted by γ {R2} (G). Roman {2}-domination is a generalization of Roman domination that has also studied by Henning and Klostermeyer [8] with the name Italian domination.…”

Section: Introductionmentioning

confidence: 99%

Total Roman {3}-domination in Graphs

2020

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For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.

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“…Chellali et al [7] initiated the study of a variant of Roman dominating functions, call Roman {2}-dominating functions, which is the same as W2DFs. Henning et al [8] named the new concept Italian domination. The definition of the new domination is the following.…”

Section: Introductionmentioning

confidence: 99%

More Results on Italian Domination in Cn□Cm

et al. 2020

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Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as γ I . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained γ I ( C n □ C 3 ) and γ I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained γ I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m ≥ 6 . For n ≡ 0 ( mod 3 ) , m ≡ 0 ( mod 3 ) , we obtain γ I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of γ I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l ≥ 1 ) , γ r 2 ( C n □ C m ) = m n 3 , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that C n □ C m is an example of a graph class for which γ I = γ r 2 , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394–2400.

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Italian domination in trees

Cited by 80 publications

References 6 publications

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

Total Roman {3}-domination in Graphs

More Results on Italian Domination in Cn□Cm

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