6.36. Let G = (V,E) be a graph with no isolated nodes and consider thepolytope P defined by the system of inequalitiesxu xv ≤1, for every edge uvЄ E(6.23)บx, > 0, for every node v Є V.Let be a 0-1 vector in P. Then the set of nodes vЄ V such that x,, = 1is a stable set of G, that is, no two nodes in the set are adjacent in G. Theconvex hull of the incidence vectors of the stable sets of G is called thestable set polytope of G, and denoted by S(G). Clearly, S(G) = P1. Foreach of the following classes of valid inequalities for S(G), give a cutting-plane proof starting from the system (6.23).(i) Let C CV be the node set of an odd circuit in G. Then the inequalityx(C) ≤ (|C| − 1)/2 is valid for S(G).-SEPARATION AND OPTIMIZATION237(ii) The graph illustrated in Figure 6.13 is called a 5-wheel. If W is the nodeset of a 5-wheel in G and r is the center node, then 2x, +x(W \ {r}) ≤ 2is valid for S(G).(iii) A set of nodes K CV is called a clique of G if each pair of nodes in Kare adjacent in G. For any clique K, the inequality (K) ≤ 1 is valid forS(G).

Step 1:To prove each of the classes of valid inequalities for the stable set polytope S(G) starting…

Answered: 6.36. Let G = (V,E) be a graph with no…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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