Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph btained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) + e)}.
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- Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced subgraph isomorphic to k1,r+1 has a 1-factor. Show that this is not true if you replace r connected by r edge connectedLet G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible number of edges in G? Show that this bound is sharp.