Volume 54, Number 4, July-August 2020
|Page(s)||1027 - 1040|
|Published online||20 May 2020|
Global defensive k-alliances in directed graphs: combinatorial and computational issues
Department of Mathematics, University of Mazandaran, Babolsar, Iran
2 Departamento de Matemáticas, Universidad de Cádiz, Algeciras, Spain
* Corresponding author: email@example.com
Accepted: 2 May 2019
In this paper we define the global defensive k-alliance (number) in a digraph D, and give several bounds on this parameter with characterizations of all digraphs attaining the bounds. In particular, for the case k = −1, we give a lower (an upper) bound on this parameter for directed trees (rooted trees). Moreover, the characterization of all directed trees (rooted trees) for which the equality holds is given. Finally, we show that the problem of finding the global defensive k-alliance number of a digraph is NP-hard for any suitable non-negative value of k, and in contrast with it, we also show that finding a minimum global defensive (−1)-alliance for any rooted tree is polynomial-time solvable.
Mathematics Subject Classification: 05C20 / 05C69
Key words: Global defensive k-alliance number / dominating set / k-order-sum number / rooted tree / directed tree
© EDP Sciences, ROADEF, SMAI 2020
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