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Presentation1 - The Evolution of a Hard Graph Theory...

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Click to edit Master subtitle style 7/15/11 The Evolution of a Hard Graph Theory Problem – 11

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7/15/11 Many real world problems involve a collection of entities (e.g., individuals, businesses, or countries) that compete for each others resources. A group of these entities can reach mutual agreements, or pacts, to defend themselves against a 22
7/15/11 Throughout, assume G = (V, E) is a connected graph. For a set S & V, a vertex in N[S]–S w 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. N[S]–S is sometimes called the "boundary" of S. 33

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7/15/11 There are several ways we can define these sets. We will discuss two. Secure Sets and Defensive Alliances . 44
7/15/11 Defensive Alliances A set S & V is a defensive alliance when every vertex in S has, with itself, as many neighbors in the set as outside the set – Every vertex in the alliance can be "protected." Formally, Definition: S V is a defensive alliance 55

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7/15/11 Defensive alliances have been used to model a variety of real world applications and requires only that each individual vertex in a set S be defendable . The defensive alliance number, da(G), satisfies & (&(G)+1)/2g ≤ da(G) 66
7/15/11 Secure Sets Any and all vertices in S can be attacked simultaneously. An attack on S is an assignment for each vertex in N[S]–S to attack exactly one of its neighbors in S. 77

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7/15/11 For a given attack and defense, a single vertex v ± S is defended if the number of defenders assigned to v is at least that of the number of attackers assigned to v. An attack is defendable if there is a defense in which all vertices of S are 88
Informally, a group (set) is secure when it can successfully defend itself against any possible take-over or attack. Definition:

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This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.

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Presentation1 - The Evolution of a Hard Graph Theory...

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