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SecureSets - The Evolution of a Hard Graph Theory Problem...

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The Evolution of a Hard Graph Theory Problem – Secure Sets Ron Dutton Computer Science University of Central Florida A Secure Set of a graph is a set S of vertices with the property that for every subset X of S, N[X] S contains as many vertices as there are neighbors of X not in S. This is a special kind of Defensive Alliance, which only requires the property to hold for all singleton subsets of S. The question of whether a graph G has a Defensive Alliance of no more than k vertices, for arbitrary G and k, is known to be NP-Complete. The similar question for Secure Sets seems to be much harder than any NP- Complete problem. In fact, if P ≠ NP, it is probably not even in the set NP. On the other hand, if P = NP, it can be solved by polynomial algorithms. How this all comes about and at least part of what it means will be the topic of this presentation. 1. Introduction Many real world problems involve a collection of entities, for example, individuals, businesses, or countries, that compete for resources. A group of these entities can reach mutual agreements, or pacts, to simultaneously attack, or defend against, a common adversary. Informally, a group is secure when they can successfully defend themselves against any possible take-over or attack. Under certain conditions, this type of problem can be modeled as a graph with a group reaching an accord corresponding to a Secure Set in the graph. To date the complexity of determining when a graph has a secure set of some given size or, even for a given a set, determining whether it is or is not secure is unknown. Most of the literature on secure sets attempts to bound the size of a minimum secure set either for special classes of graphs or in terms of other graphical invariants. Section 3 shows formally that determining if a give set of vertices is a secure set is a difficult problem, which indicates the former problem may be even more difficult. Section 4 presents results that may prove helpful in developing algorithms and heuristics for these and related problems. At the least, these results provide insight into the structure and properties of secure sets. 1
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2. Background Basic graph theory terminology can be found in [ ], while that involving complexity theory can be found in Garey and Johnson [ ]. Throughout, assume G = (V, E) is a connected graph. For a set S V, a vertex in N[S]–S V–S can "attack" any one of it's neighbors in S, while a vertex in S can "defend" itself or any one of its neighbors in S. An attack on S is an assignment for each vertex in N[S]–S to attack exactly one of its neighbors in S. A defense
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