Problem 2. Consider a graph G = (V,E) where |V|=n. 2(a) What is the total number...

Problem 2. Consider a graph G = (V,E) where |V|=n.

2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates.

2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t?

2(c) What is the total number of possible cycles of any length in G from a given starting vertex s?

Expert Solution

Problem 2. Consider a graph G = (V,E) where |V|=n.

2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates.

2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t?

2(c) What is the total number of possible cycles of any length in G from a given starting vertex s?

Colby Messinger answered 2 years ago

# Problem Description Given a directed graph G = (V, E), find the number of connected...

# Problem Description Given a directed graph G = (V, E), find the number of connected components in G. # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives two numbers `n` and `m`, describing the number of vertices and edges. Each of the following lines contains two numbers `a` and `b` meaning there is an edge (a,b) belong to E. All the numbers in a line are separated by...

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3...

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. a) Draw a maximal plane graphs on six vertices. b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon...

Consider an unweighted, undirected graph G = <V, E>. The neighbourhood of a node v ∈...

Consider an unweighted, undirected graph G = <V, E>. The neighbourhood of a node v ∈ V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v). (a) Design an algorithm that returns a list containing the neighbourhood degree for each node v ∈...

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V...

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such...

Prove that if G is a simple graph with |V (G)| = n even, where δ(G)...

Prove that if G is a simple graph with |V (G)| = n even, where δ(G) ≥ n 2 + 1, then G has a 3-regular spanning subgraph.

If G = (V, E) is a graph and x ∈ V , let G \...

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned...

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V1 and V2, such that every edge joins a vertex in V1 with a vertex in V2. This means no edges are within V1 or V2 (or symbolically: ∀u, v ∈ V1, {u,v} ∉ E and ∀u,v ∈ V2, {u,v} ∉ E). 8(a) Show that the complete graph K2 is a bipartite graph. 8(b) Prove that no complete graph Kn,...

(1) Run Prim's algorithm and Kruskal's algorithm for the following graph: G(V, E) where V =...

(1) Run Prim's algorithm and Kruskal's algorithm for the following graph: G(V, E) where V = {A, B, C, D, E} E = { {A,B}, {B,C}, {C, A}. {B, D}, {B, E}, {D, E}} use edge weights: w({A, B} = 1), w({B, C}) = 2, w({C, A}) = 3, w({B, D}) = 4, w({B, E}) = 5, w({D, E}) = 6. (2) Can the maximum weight edge be in every minimum spanning tree of a graph?

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has...

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has degree at least 4 and u is in V , then there are at least 64 distinct paths in G that start at u.

# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0...

# Problem Description Given a directed graph G = (V,E) with edge length l(e) > 0 for any e in E, and a source vertex s. Use Dijkstra’s algorithm to calculate distance(s,v) for all of the vertices v in V. (You can implement your own priority queue or use the build-in function for C++/Python) # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives three numbers `n`,`m` and `s`(1 <=...

Question