Let DIST (u, v) denote the distance between vertex u and v. It is well known that distances in graphs satisfy the triangle inequality. That is, for any three vertices u, v, w, DIST (u, v) ≤ DIST (u, w) + DIST (w, v). Let D∗ denote the distance between the two farthest nodes in G. Show that for any vertex s    D∗  ≤  2 max DIST (s, v).

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.

Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is

1. Run DFS-with-timing on this graph G: give the pre and post number of each vertex. Whenever there is a choice of vertices to explore, always pick the one that is alphabetically first. 2. Draw the meta-graph of G. 3. What is the minimum number of edges you must add to G to make it strongly connected (i.e., it consists of a single connected component after adding these edges)? Give such a set of edges. b.

Let G be a graph, where each edge has a weight. A spanning tree is a set of edges that connects all the vertices together, so that there exists a path between any pair of vertices in the graph. A minimum-weight spanning tree is a spanning tree whose sum of edge weights is as small as possible. Last week we saw how Kruskal's Algorithm can be applied to any graph to generate a minimum-weight spanning tree. In this question, you will apply Prim's Algorithm on the graph below. You must start with vertex A. H 4 4 1 3 J 2 C 10 4 8 B 9 F 18 8 There are nine edges in the spanning tree produced by Prim's Algorithm, including AB, BC, and IJ. Determine the exact order in which these nine edges are added to form the minimum-weight spanning tree. 3.

Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple (no repeated vertices).

Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.

Consider an undirected graph on 8 vertices, with 12 edges given as shown below: Q2.1 Give the result of running Kruskal's algorithm on this edge sequence (specify the order in which the edges are selected). Q2.2 For the same graph, exhibit a cut that certifies that the edge ry is in the minimum spanning tree. Your answer should be in the form E(S, V/S) for some vertex set S. Specifically, you should find S.

Construct all regular graphs that exist for the vertex set V = {1, 2, 3, 4}. For example, G = (V,E0) where E0 = {} is one possible graph

Write a function that returns true (1) if there is an edge between two vertices u and v in an adjacency-listed directed graph.