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?

Solutions

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 1 year ago

Next > < Previous

Related Solutions

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.

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.

Let G be a simple undirected graph with n vertices where n is an even number....

Let G be a simple undirected graph with n vertices where n is an even number. Prove that G contains a triangle if it has at least (n^2 / 4) + 1 edges using mathematical induction.

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each...

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each edge labeled from the set {0,1}. Describe and analyze the worst-case time complexity of an efficient algorithm to find any cycle consisting of edges whose labels alternate 0,1.

Problem for submission: For which positive integers k can a simple graph G = (V, E)...

Problem for submission: For which positive integers k can a simple graph G = (V, E) be constructed such that: G has k vertexes, that is, |V | = k, G is bipartite, and its complement G is bipartite? Prove your answer is correct Please show and explain your full proof.

Subjects