Question: Let G be an undirected graph with vertex set V. Define the relation R on V by v R w if v = w or

Let G be an undirected graph with vertex set V. Define the relation R on V by v R w if v = w or if there is a path from v to w (or from w to v since G is undirected),
(a) Prove that is an equivalence relation on V.
(b) What can we say about the associated partition?

Step by Step Solution

★★★★★

3.35 Rating (164 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a For each v V v v so vRv If vRw then there is a path from v to w Since the graph ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (7900).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

(a) Let G be an undirected graph with n vertices. If G is isomorphic to its own complement , how many edges must G have? (Such a graph is called self-complementary.) (b) Find an example of a...

(a) Let G = (V, E) be an undirected graph, with G1 = (V1, E1) a subgraph of G. Under what condition(s) is G1 not an induced subgraph of G? (b) For the graph G in Fig. 11.27(a), find a subgraph that...

Let G = (V, E) be an undirected graph with subset I of V an independent set. For each a I and each Hamilton cycle C for G, there will be deg (a) - 2 edges in E that are incident with a and not in C....

Exercise 3. Let G be a connected graph, with vertex set V. The distance between two vertices u and v, denoted dist(u, v), is defined as the minimal length of a path joining u to v. Show that the...

Let G = ( V , E ) be an undirected graph with vertex set V and edge set E . A 3 - coloring of G is a map \ chi : V - > { R , B , Y } such that if { x , y } in E then \ chi ( x ) = \ chi ( y ) . (...

Let G = G(V,E) be an arbitrary, connected, undirected graph with vertex set V and edge set E. Assume that each edge e 2 E has a distinct positive length `e, i.e., if e 6= e0 then `e 6= `e0. Define...

Urgent MAT 1348 Final Exam 11 Q18. (6 points) Recall that, in graph, a walk of length n 2 0 from vertex a to b is an al- ternating sequence of vertices and edges voelve2veg .. . Un-1enUn, where vo =...

6.3 Graph kernel. Let G = (V; E) be an undirected graph with vertex set V and edge set E. V could represent a set of documents or biosequences and E the set of connections between them. Let w[e] 2 R...

Algorithmic: 1 . Given the following directed graph: A . Find the shortest path length from vertex A to any other vertex in the graph by running the BFS algorithm. B . Extend the BFS algorithm so...

If G = (V, E) is a loop-free connected undirected graph and a, b V, then we define the distance from a to b (or from b to a), denoted d(a, b), as the length of a shortest path (in G) connecting a and...

Last year, 35% of people who downloaded a free app (application) for their mobile electronics device deleted the downloaded app within 6 months of the download. Acme games recently conducted a survey...

I Cover IT makes iPhone and iPad covers. The employee payroll averages $300 000 per week. I Cover IT would like to extend the pay cycle to fortnightly. The manufacturing workers union argues the...

Identify three forms of certification available to managerial accountants.

Is Jill Johnson correct when she says the following? I am currently producing 10 000 pizzas per month at a total cost of $50 000. If I produce 10 001 pizzas, my total cost will rise to $50,011....

Find the components of the vectors u, v, u + v, and u - v, where u and v are as shown in Figure 1 .23. 60 -1 30

In Figure 1 .24, A, B, C, D, E, and F are the vertices of a regular hexagon centered at the origin. Express each of the following vectors in terms of a=01 and b = 08. (a) AB (c) AD (e) AC (b) BC (d)...

In Exercises 1 5 and 1 6, simplify the given vector expression. Indicate which properties in Theorem 1. 1 you use. a. 2(a - 3b) + 3 (2b + a) b. - 3(a - c) + 2(a + 2b) + 3(c - b)

Suppose you want to hedge a $330 million bond portfolio with a duration of 5.5 years using 10-year Treasury note futures with a duration of 7 years, a futures price of 107, and 9 months to...

Can you explain it step by step please

Explain this statement: "It may be better to target one competitor in many countries than many competitors in a few countries." Why is this the case? Explain this statement: "It may be better to...