Form of the smallest vertex cover in a bipartite graph I'm trying to write a proof of Konig's theor

dream13rxs 2022-07-14 Answered
Form of the smallest vertex cover in a bipartite graph
I'm trying to write a proof of Konig's theorem using Menger's theorem. However, I got stuck along the way. In order to move forward I'd (apparently) need to show the following fact.
Let G be a bipartite graph with partite sets X and Y. The smallest vertex cover of G is of the form ( X A ) N ( A ) for some A X.
I tried doing a proof by contradiction but to no avail. Maybe the thesis I'm trying to show is false in the first place? Any help would be greatly appreciated.
You can still ask an expert for help

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

zlepljalz2
Answered 2022-07-15 Author has 22 answers
Step 1
Let U be any vertex cover; let A = X U (the smallest set for which U contains X A).
For every vertex b N ( A ), there is some edge ab with a A which needs to be covered by U somehow. By construction a U, so ab is not covered by a. Therefore ab must be covered by b: we must have b U. Therefore N ( A ) U.
Step 2
We have shown that U contains ( X A ) N ( A ). We can check that ( X A ) N ( A ) is a vertex cover all by itself. Therefore if U is a minimal vertex cover (if it has no proper subset which is a vertex cover) we must have U = ( X A ) N ( A ). In particular, this is true of the smallest vertex cover.
I admit I do not see how you need this fact to prove Konig's theorem from Menger's theorem. Neither one deals in N(A) as a concept. Maybe you are trying to prove Hall's theorem from Konig's theorem?

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-08-15
How many elements are in the set { 0, { { 0 } }?
asked 2021-08-02
Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols:
asked 2021-07-28

Let A, B, and C be sets. Show that (AB)C=(AC)(BC)
image

asked 2021-08-18
Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b)R if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
asked 2022-08-19
Given the statementS: ''If you are a computer science student, then you know either discrete math or Java'',which one is the negation of the statement S ?1. you are not a computer science student, but you know either discrete math or Java2. If you are a computer science student, then you know neither discrete math nor Java3. You are a computer science student, but you know neither discrete math nor Java4. If you are not a computer science student, then you know neither discrete math nor Java
asked 2022-09-07
Discrete Math Help with a Proof
I need help to prove the following: Let a, b, and c be any integers. If a∣b, then a∣bc
asked 2022-07-04
What would the negation of these two statements be?
I need to negate these two statements and I believe that I have the quantifiers correct, but I'm not completely sure how to negate the math statements. I think I would keep the equations before the equal signs the same but I'm still unsure how to negate the last parts.
Here are the statements I want to negate:
Statement 1:
x , y R 0 , 9 x 2 + y 2 3 x + y .
Statement 2:
x R 0 , 25 x 2 + 9 = 5 x + 3.

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question