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.
Mots-clés : Global defensive $$-alliance number, dominating set, $$-order-sum number, rooted tree, directed tree
@article{RO_2020__54_4_1027_0,
author = {Mojdeh, Doost Ali and Samadi, Babak and Yero, Ismael G.},
title = {Global defensive $k$-alliances in directed graphs: combinatorial and computational issues},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {1027--1040},
publisher = {EDP-Sciences},
volume = {54},
number = {4},
year = {2020},
doi = {10.1051/ro/2019049},
mrnumber = {4100699},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2019049/}
}
TY - JOUR AU - Mojdeh, Doost Ali AU - Samadi, Babak AU - Yero, Ismael G. TI - Global defensive $k$-alliances in directed graphs: combinatorial and computational issues JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2020 SP - 1027 EP - 1040 VL - 54 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2019049/ DO - 10.1051/ro/2019049 LA - en ID - RO_2020__54_4_1027_0 ER -
%0 Journal Article %A Mojdeh, Doost Ali %A Samadi, Babak %A Yero, Ismael G. %T Global defensive $k$-alliances in directed graphs: combinatorial and computational issues %J RAIRO - Operations Research - Recherche Opérationnelle %D 2020 %P 1027-1040 %V 54 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2019049/ %R 10.1051/ro/2019049 %G en %F RO_2020__54_4_1027_0
Mojdeh, Doost Ali; Samadi, Babak; Yero, Ismael G. Global defensive $k$-alliances in directed graphs: combinatorial and computational issues. RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 4, pp. 1027-1040. doi : 10.1051/ro/2019049. http://www.numdam.org/articles/10.1051/ro/2019049/
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