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}

max edge distance Simplification key Figure 4-1: A sample process for the Douglas-Peucker algorithm The Douglas-Peucker algorithm is for the selection of representative points to simplify a curve composed of line segments. It uses a point-to-edge distance tolerance. The algorithm starts with a crude simplification that is the single edge joining the first and last vertices of the original polyline. It then computes the perpendicular distance of all intermediate vertices to that edge. The vertex that is furthest away from that edge, and that has a computed distance that is larger than a specified tolerance, will be marked as a key and added to the simplification. This process will recurse for each edge in the current simplification until all vertices of the original polyline are within tolerance of the simplification results. This process is illustrated in Figure 4-1. (1) Given three points (xp, Yp), (Xa, Ya), (Xp,Yb), show a detailed process to compute the perpendicular distance from p…

*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).

Additional Topics: Topological Sort A strict order relation can be represented by a directed acyclic graph (or DAG), which is a directed graph with no cycles. For example, the following directed graph is a DAG: A topological sort of a DAG is an ordering of the vertices that is consistent with the edges of the graph. That is, if there is an edge (u, v) in the graph, then u should appear before v in the topological sort. For example, 1,2,3 and 1,3, 2 are topological sorts of the DAG shown above, but 2,1,3 is not a topological sort because 2 cannot be listed before 1. 7. Given the following DAG G representing UIC's Computer Science courses and prerequisites (note that edges implied by the transitive property are omitted): CS 261 CS 362 CS 141 CS 11 CS 211 Cs 361 CS 401 CS 151 CS 251 CS 342 Cs 301 Answer the following questions: a. Give two different topological sorts of G. b. Is each of the following orderings a topological sort of G? Justify your answer. CS111, CS151, CS301, CS251,…

Question P 3.Full explain this question and text typing work only     We should answer our question within 2 hours takes more time then we will reduce Rating Dont ignore this line  Fast

4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem   You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…

The number of elements in the adjacency matrix of a graph having 7 vertices is: O a. 36 O b. 7 O c. 49 O d. 14

A simple directed graph with vertices A,B,C,D,E,F,G has adjacency matrix

Weighted Graph Applications Demonstration Java Data Structures. Figure 29.23 illustrates a weighted graph with 6 vertices 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). 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). You can therefore attach proper content files with dot txt, png, jpg or jpeg extensions  Be sure the final trees or path lists are clearly visible in your solution.  You don't need to show the solution development or progress, just the result.

an undirected graph is called complete if every shares an edge with every other vertex.