Corollary 5.6. (i) If G is a connected simple planar graph with (n> 3) vertices and m edges, then m < 3n (ii) If in addition, G has no triangles, then m < 2n – 4. 6. |
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- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1An undirected weighted graph G is given below: 3 .2 5 Figure 16: An undirected weighted graph has 6 vertices, a through f, and 9 edges. Verter d is on the left. Verter f is above and to the right of verter d. Verter e is below and to the right of verter f, but above vertez d. Verter e is belou and to the right of vertez e. Vertez a is above verter e and to the right of verter c. Verter 6 is below and to the right of vertez a, but above vertez e. The edges between the vertices and their weight are as follous: d and f, 1; d and e, 4: f and e, 2; e and a, 2: f and a, 3; e and c, 5; e and a, 7; e and b, 5; and a and b, 6. Use Prim's algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree. What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge {d, e} is in the span- ning tree? How would you generalize this idea? Suppose…5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)Give an example of a graph that has all 3 of the following properties. (Note that you need to give a single graph as the answer.) (i) It is connected (ii) It has one articulation point. (iii) The graph needs at least 4 colors for a valid vertex coloring (iv) The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.Recall the complete bipartite graph Km.n. a) Draw K5,3. b) Determine the number of vertices in Km.n. c) Determine the number of edges in Km,n. d) The complement of a graph G is the graph G with same vertex set; but complement edge set. Draw K4,3. e) What kind of graph isKm.n.Let V= {cities of Metro Manila} and E = {(x; y) | x and y are adjacent cities in Metro Manila.} (a) Draw the graph G defined by G = (V; E). You may use initials to name a vertex representing a city. (b) Apply the Four-Color Theorem to determine the chromatic number of the vertex coloring for G.3. From the graph above determine the vertex sequence of the shortest path connecting the following pairs of vertex and give each length: a. V & W b. U & Y c. U & X d. S & V e. S & Z 4. For each pair of vertex in no. 3 give the vertex sequence of the longest path connecting them that repeat no edges. Is there a longest path connecting them?Draw the following graphs: i. K6 ii. K2,4 iii. C4, that is, the complement of the cycle graph C4 Draw the smallest (i.e. with the least number of vertices and edges) non-trivial (i.e. it has at least one edge) self-complementary graph (GG). Draw all the cubic (i.e. the degree of each vertex equals 3) graphs with at most 8 vertices.An undirected weighted graph G is given below: Figure 16: An undirected weighted graph has 6 vertices, a through f, and 9 edges. Vertex d is on the left. Vertex f is above and to the right of vertex d. Vertex e is below and to the right of vertex f, but above vertex d. Vertex c is below and to the right of vertex e. Vertex a is above vertex e and to the right of vertex c. Vertex b is below and to the right of vertex a, but above vertex c. The edges between the vertices and their weight are as follows: d and f, 1; d and e, 4; f and e, 2; e and a, 2; f and a, 3; e and c, 5; c and a, 7; c and b, 5; and a and b, 6.(a) Use Prim’s algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree.(b) What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge {d, e} is in the spanning tree?(c) How would you generalize this idea? Suppose…3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k. Show that the Strongly Independent Set Problem is NP-Complete.4. Let G=(V, E) be the following undirected graph: V = {1, 2, 3, 4, 5, 6, 7, 8} E = {(1, 7) (1, 4), (1, 6), (6, 4), (7, 5), (2, 5), (2, 4), (4, 3), (4, 8), (8, 3), (3, 5)}. a. Draw G. b. Is G connected? c. Give the adjacency matrix for the graph G given above. d. Determine whether the graph has a Hamiltonian cycle. If the graph does have a Hamiltonian cycle, specify the nodes of one. If it does not, prove that it does not have one. e. Determine whether the graph has a Euler cycle. If the graph does have a Euler cycle, specify the nodes of one. If it does not, prove that it does not have one.SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. 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