8) Solve the all-pairs shortest paths problem on the following directed graph using a dynamic-programming algorithm. All steps of the solution are written in the details. 6 8 -5
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- . Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is Hamiltonian. For each of the following, draw an Eulerian graph that satisfies the conditions, orprove that no such graph exists.a) An even number of vertices, an even number of edges.b) An even number of vertices, an odd number of edges.c) An odd number of vertices, an even number of edges.d) An odd number of vertices, an odd number of edges.(a) Draw a connected 3-regular graph with 6 vertices and no loops (b) Show that every 3-regular graph must have an even number of vertices. (c) Find two non-isomorphic connected 3-regular graphs with 6 vertices.
- 4. This problem is about graphs. In each of the following cases, either draw a graph with the stated property, or prove that no such graph exists. (For the purpose of this problem, a “graph" must have no loops or multiple edges.) (a) A graph on 12 vertices in which every vertex has degree 4. (b) A graph on 45 vertices in which 22 vertices have degree 8 and 23 vertices have degree 7. (c) A graph on 10 vertices in which every vertex has degree 3 and the shortest cycle subgraph has length 5. (d) A graph on 13 vertices in which every vertex has degree at least 7 and there are no cycle subgraphs of length 3.Choose the correct statement. [Choose the most appropriate answer] O None of the option is correct O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an odd degree. O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an even degree. O A graph G has an Euler circuit if and only if G is connected and every vertex of G has an equal degree.{01] Assume n > 3. For which values of n do these graphs have an Euler circuit? (a) Complete graph Kn. (b) Cycle graph Cn. (c) Wheel graph Wn as defined in the lecture. (d) Complete bipartite graph Kn,n-
- 2. [20 points][MID] 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 e(r) such that 1 < e{u) < k and c(u) # c(v) for every edge (u, v) in E. In other words you want to color each vertex with one of the k colors you have and no two adjacent vertices can have the same color. For example, the following graph can be 3-colored using the following color assignments: a=1,b=2,c=1,d%32,e=3,f=2.g33 a---b---c---g d Formulate the graph k-coloring problem as an evolutionary optimization. You may use a vector of integer representation, OR any representation that you think is more appropriate. you should specify: • A representation. • itness function. Give 3 examples of individuals and their fitness values if you are solving the above example. A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn…Give an example of each of the following: (4)i) a nonsimple plane graph with its dual a simple graphii) a self-dual plane graphiii) a plane graph whose each face is bounded by a triangleiv) a plane graph on 8 vertices with its dual having just one vertexIn Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
- a) List all of the paths from 1 to 4 in the graph S with a length of 5. b) The adjacent matrix for the graph R U SYou have discovered an old paper on graph theory that discusses the viscosity of a graph (which for all you know, is something completely made up by the author). A theorem in the paper claims that "if a graph satisfies condition (V), then the graph is viscous." Which of the following are equivalent ways of stating this claim? Which are equivalent to the converse of the claim? 1. Satisfying condition (V) is a necessary condition for a graph to be viscous. ? ? ? ? ? 2. For a graph to be viscous, it is sufficient for it to satisfy condition (V). 3. A graph is viscous only if it satisfies condition (V). 4. Satisfying condition (V) is a sufficient condition for a graph to be viscous. 5. Only viscous graphs satisfy condition (V).a. State the conditions for an Euler Path and the conditions for an Euler Cycle. b. Does the graph B below have an Euler Path, and Euler Cycle, or Neither? 6. For the graph B above, find the following: a. the induced subgraph on (a,b,h,i,f} b. a subgraph that is neither spanning or induced on the vertices {c,d,g,f,e} 5.