Given a graph data structure: G = (V,E) where, V = {A, B, C, D, E } E = { (A,B), (A,D), (B,D), (B,C), (D,E) } a) Draw the graph G. b) Draw all spanning trees of the graph G, which are also “linear” (each graph node has no more than two neighbors).
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- A graph G=(v, e) is a _______ graph, if we can partition V=L u R, s.t. all edges in E go between L and R.1. Construct a simple graph that is a forest with vertices M, N, O, P, Q, R such that the degree of O is 2 and there are 2 components. What is the edge set? 2. Construct a simple graph that is a tree with vertices P,Q,R,S,T,U such that the degree of U is 4. What is the edge set?Discrete Mathmatics The graph intersection of a collection of sets A1, A2, · · · , An is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph for of the following collection of sets. A1 = {0, 2, 4, 6, 8} A2 = {0, 1, 2, 3, 4} A3 = {1, 3, 5, 7, 9} A4 = {5, 6, 7, 8, 9} A5 = {0, 1, 8, 9}
- Throughout, a graph is given as input as an adjacency list. That is, G is a dictionary where the keysare vertices, and for a vertex v,G[v] = [u such that there is an edge going from v to u].In the case that G is undirected, for every edge u − v, v is in G[u] and u is in G[v] 5. Write the full pseudocode for the following problem using BFS/DFS.Input: An undirected graph G that’s not necessarily connected.Output: Is G bipartite? That is, can the vertices of G be partitioned into two sets X, and Y such that thereare no edges within X and Y ?Fill in the missing entries to complete the adjacency list representation of the given graph. V = {1, 2, 3, 4, 5) E = {(1, 2), (2, 3), (4, 3}, {5, 4}} (a): Ex: 4 (b): (c): (d): 1 2 3 4 5 (a) 1 3 2 (b) 3 (c)Y3
- In this task you will work with an undirected Graph G = {V, E}, where V = {f,p,s,b,l,j,t,c,d} and E = {(1,2), (1,3), (3,8), (4,8), (8,9), (1,7), (2,6), (2,3), (5,6), (6,7), (7,9), (8,1)}. Assume that the nodes are stored in an indexed linear structure (e.g., an array or a vector) numbered consecutively from 1 (node f) to 9 (node d). 1. Type up a tgf representation of the graph 2. Find a path from node l to node b with length 8, that passes through every vertex of the graph. List the nodes of that path. 3. The graph G contains cycles. What is the smallest number of vertices to remove in order to break all cycles?Assume a graph G is simple (ie. no self loop or parallel edges) .Let v be any vertex in the graph. Let boolean[] marked be initalized to all false. Consider: boolean dfs(Graph G, int v) { marked[v] = true; for (int w : G.adj(v)) { if (w == u) continue; if (marked[w]) return true; if (dfs (G, v, w)) return true; } return false; } If the call dfs(v) returns true, then: a. The graph has cyclesb. The graph is bipartitec. The graph is connectedData Structures Weighted Graph Applications Demonstration Look at Figure 29.23 which illustrates a weighted graph with 6 verticies and 8 weighted edges. Simply provide: Minimal Spanning Tree as an illustration or a textual list of edges (in our standard vertex order). Single-Source Shortest Path route from vertex 0 to the other 5 (described as one path/route for each). You may draw the two solutions and attach the illustration or describe them in text (a list of edges for the one and the vertex to vertex path the other). Be sure the final trees or path lists are clearly visible in your solution.
- Consider the graph in the figure below. E B F G AZ. с I H A a) List all the bridges in this graph. Enter your answers in alphabetical order separated by commas but no spaces. b) If you remove all of the bridges from this graph, how many components will the resulting graph have? c) What is the length of the shortest path from C to F? Length: 2 d) What is the length of the longest path from I to J? Length: 5 Check J ✔Path: C,I,F Path: I,G,F,I,H,J X X*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).Hippity hoppity, abolish loopity def frog_collision_time(frog1, frog2): A frog hopping along on the infinite two-dimensional lattice grid of integers is represented as a 4- tuple of the form (sx, sy, dx, dy) where (sx, sy) is its starting position at time zero, and (dx, dy) is its constant direction vector for each hop. Time advances in discrete integer steps 0, 1, 2, 3, ... so that each frog makes one hop at every tick of the clock. At time t, the position of that frog is given by the formula (sx+t*dx, sy+t*dy) that can be nimbly evaluated for any t. Given two frogs frog1 and frog2 that are guaranteed to initially stand on different squares, return the time when both frogs hop into the same square. If these two frogs never simultaneously arrive at the same square, return None. This function should not contain any loops whatsoever. The result should be calculated using conditional statements and integer arithmetic. Perhaps the best way to get cracking is to first solve a simpler…