Secure sets and their expansion in cubic graphs
AbstractConsider a graph whose vertices play the role of members of the opposing groups. The edge between two vertices means that these vertices may defend or attack each other. At one time, any attacker may attack only one vertex. Similarly, any defender fights for itself or helps exactly one of its neighbours. If we have a set of defenders that can repel any attack, then we say that the set is secure. Moreover, it is strong if it is also prepared for a raid of one additional foe who can strike anywhere. We show that almost any cubic graph of order n has a minimum strong secure set of cardinality less or equal to n/2 + 1. Moreover, we examine the possibility of an expansion of secure sets and strong secure sets.
1987 ◽
Vol 30
(2)
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pp. 193-199
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2010 ◽
Vol 62
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pp. 355-381
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1987 ◽
Vol 36
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pp. 441-447
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1970 ◽
Vol 11
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pp. 207-215
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2012 ◽
Vol 22
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pp. 765-778
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2015 ◽
Vol Vol. 17 no. 1
(Graph Theory)
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2014 ◽
Vol Vol. 16 no. 3
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1992 ◽
Vol 1
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pp. 371-381
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2004 ◽
Vol 76
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pp. 345-356
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