Kharazi, H., Mosleh Tehrani, A. (2019). On the defensive alliances in graph. Transactions on Combinatorics, 8(1), 1-14. doi: 10.22108/toc.2018.50156.1396
Hasan Kharazi; Alireza Mosleh Tehrani. "On the defensive alliances in graph". Transactions on Combinatorics, 8, 1, 2019, 1-14. doi: 10.22108/toc.2018.50156.1396
Kharazi, H., Mosleh Tehrani, A. (2019). 'On the defensive alliances in graph', Transactions on Combinatorics, 8(1), pp. 1-14. doi: 10.22108/toc.2018.50156.1396
Kharazi, H., Mosleh Tehrani, A. On the defensive alliances in graph. Transactions on Combinatorics, 2019; 8(1): 1-14. doi: 10.22108/toc.2018.50156.1396
Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.
Abstract
Let $ G = (V,E) $ be a graph. We say that $ S \subseteq V $ is a defensive alliance if for every $ u \in S $, the number of neighbors $ u $ has in $ S $ plus one (counting $ u $) is at least as large as the number of neighbors it has outside $ S $. Then, for every vertex $ u $ in a defensive alliance $ S $, any attack on a single vertex by the neighbors of $ u $ in $ V-S $ can be thwarted by the neighbors of $ u $ in $ S $ and $ u $ itself. In this paper, we study alliances that are containing a given vertex $ u $ and study their mathematical properties.
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