Question: (a) Let G = (V, E) be a directed graph or multigraph with no isolated vertices. Prove that G has a directed Euler circuit if

(a) Let G = (V, E) be a directed graph or multigraph with no isolated vertices. Prove that G has a directed Euler circuit if and only if G is connected and od(v) = id(v) for all v ∈ V.
(b) A directed graph is called strongly connected if there is a directed path from a to b for all vertices a, b, where a ≠ b. Prove that if a directed graph has a directed Euler circuit, then it is strongly connected. Is the converse true?

Step by Step Solution

★★★★★

3.30 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a If G V E has a directed Euler circuit then for all x y V there is a directed trail from x to y that part of the directed Euler circuit from x to y T... 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 (8138).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

If G = (V, E) is a directed graph or multigraph with no isolated vertices, prove that G has a directed Euler trail if and only if (i) G is connected; (ii) od(v) = id(v) for all but two vertices x, y...

Show that a directed multi graph having no isolated vertices has an Euler path but not an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex...

If G = (V, E) is an undirected graph, a subset D of V is called a dominating set if for all v V, either v D or v is adjacent to a vertex in D. If D is a dominating set and no proper subset of D has...

Let G = (V, E) be a directed graph where V = {1, 2, , n}. Given a vertex v let Sv denote the set of vertices that have a path from v (excluding v), and let Tv be the set of vertices from which there...

Let G = (V, E) be a directed graph where V = { 1,2, . . . ,n} such that n is odd; Le n = 2k+1 for some k > 0. Given a vertex v, let TO, be the set of all vertices from which there is path to v. Let...

3 Max-Flow and Graph Reliability (35 pts] Let G = (V, E) be a directed graph with two specified vertices st EV. Assume that contains at least one s-t path. A subset of edges E' CE is a separating...

Question 2. I. Let G = (V. E) be a directed graph in which each vertex u E V is labeled with a unique integer L(u) from th.2 V For each vertex u E V, let Ru be the set of vertices that are reachable...

Let G = (V, E) be a directed graph in which each vertex u V is labeled with a unique integer L(u) from the set {1, 2, ..., |V |}. For each vertex u V , let R(u) be the set of vertices that are...

Let G = (V, E) be a directed graph. We say that a set of paths are non-overlapping if they have no edges in common. For example, the set of 3 (purple, orange, and green) paths in the graph below form...

Kay terminated employment with Atria Corp after completing four years of service. Atria sponsors a 401(k) profit sharing plan (non-safe harbor) with a dollar for dollar match up to 6% of compensation...

A survey of footwear preferences of a random sample of 100 undergraduate students (50 females and 50 males) from a large university resulted in the following data. (a) Let pi, i = 1, 2, 3, 4, 5,...

4 0 3 ( b ) plans are self - directed retirement accounts set up by individuals and contributed to only by individuals. Group startsTrue or False True, unselectedFalse, unselected

On January 1, Mrs. Irene Kern leased an apartment from Colonial Court Apartments, Inc., for a one-year term. When the lease was entered into, Mrs. Kern asked for a quiet apartment, and Colonial...

Let T: V V be a linear transformation, and let {v1, . . . , vn} be a basis for V such that {T(v1), . . . , T (v)} is also a basis for V. Prove that T is invertible.

Determine whether {1, cos 2x, 3 sin2x} is linearly dependent or independent.

Let A and B be nonzero n n matrices such that A is symmetric and B is skew-symmetric. Prove that {A, B} is linearly independent.

You've observed the following returns on Crash-n-Burn Computer's stock over the past five years: 2 percent. -14 percent, 25 percent, 16 percent, and 13 percent. What was the arithmetic average return...

You would be more likely to borrow through an FRM if you believed that interest rates were going to increase. True False Question 4 The Federal Housing Administration (FHA) does not guarantee...

a company has a portfolio of stocks worth $100 million. the beta of the porfolio is 1.2. the company would like to use the CME december futures contract on the s&p 500 to change the beta of the...