Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.
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- Use symmetry to sketch the graph of xy2=1.Based on the result in Exercise 26, describe the graph of the equation x2+y2=16.Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.
- Q₁. Construct a connected graph with at least 12 vertices such that the minimal dominating set has 6 vertices and the minimum dominating set has 4 vertices.Q5. a) If G (V, E) is a connected undirected graph with |E| = 19 and deg(v) >=4 for all v ∈ V , what is the maximum value for |V|. b) What will be the number of edges in graph Q8 ? Explain with proper justification.A school consists of 6 separate buildings, represented by the vertices in the graph. There are paths between some of the buildings as shown. The graph also shows the length in feet of each path. School administrators want to cover some of these paths with roofs so that students will be able to walk between buildings without getting wet when it rains. To minimize cost, they must select paths to be covered such that the total length to be covered is as small as possible. Use Kruskal's algorithm to determine which paths to cover. Also determine the total length of pathways to be covered. OA. Click here to view figure c. OC. Click here to view figure b. A The total length of pathways to be covered is (Type a whole number.) 45 B Find a minimum spanning tree. Choose the correct answer below. 51 34 19 39 25 E 31 29 30 25 F41 (... D 35 O B. Click here to view figure d. O D. Click here to view figure a.
- Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.Show that every simple finite graph has two vertices of the same degree.