Let G = (V, E) be a graph. A subset D of V is called dominating if for all v ∈ V, either v ∈D or v is adjacent to some vertex in D. If D is a dominating set, and no proper subset of D is a dominating set, then D is called minimal. If G has no isolated vertices, prove that if D is a minimal dominating set for G, then V – D is a dominating set for G.

1.36. Show that if G is a tree then for any pair of vertices a, b there is exactly one path from a to b in G.

Consider an undirected graph G = (V, E). An independent set is a subset I CV such that for any vertices i, j e I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G.

a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ = (V, E₂) to be distinct, they need only have E₁ # E2, but they may be isomorphic.) b. In the graph below, give an example of a trail that is not a path. I Y U Z

1. Let G be a graph with 7 vertices whose weight matrix is ​​the following. Determine the Cover Tree