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Since (I) comes only in one option i.e. option <b>(A)</b>, it is correct answer. We don't need to check for other sequences.
 
Since (I) comes only in one option i.e. option <b>(A)</b>, it is correct answer. We don't need to check for other sequences.
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[[Category:Graph Theory]]
 
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[[Category: Graph Theory questions]]
 
[[Category: Graph Theory questions]]

Revision as of 17:36, 15 April 2014

The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?

I. 7, 6, 5, 4, 4, 3, 2, 1

II. 6, 6, 6, 6, 3, 3, 2, 2

III. 7, 6, 6, 4, 4, 3, 2, 2

IV. 8, 7, 7, 6, 4, 2, 1, 1


(A) I and II

(B) III and IV

(C) IV only

(D) II and IV

Solution by Happy Mittal

This can be solved using havel hakimi theorem, which says :

  1. First arrange degree sequence in decreasing order.
  2. Remove 1st vertex, and let its degree be k, then subtract 1 from next k vertices.
  3. If all vertices have degree 0, then answer is yes i.e. given degree seqeunce can be a degree sequence for a graph. If any vertex has degree < 0, then answer is no, otherwise repeat step 2.

So we check each degree sequence given in question :

  1. 7, 6, 5, 4, 4, 3, 2, 1. Here first vertex has degree 7, so remove this first vertex, and then subtract 1 from next 7 vertices, so we get 5,4,3,3,2,1,0. Then we get 3,2,2,1,0,0, then 1,1,0,0,0, and then 0,0,0,0. So answer is yes.

Since (I) comes only in one option i.e. option (A), it is correct answer. We don't need to check for other sequences.




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The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?

I. 7, 6, 5, 4, 4, 3, 2, 1

II. 6, 6, 6, 6, 3, 3, 2, 2

III. 7, 6, 6, 4, 4, 3, 2, 2

IV. 8, 7, 7, 6, 4, 2, 1, 1


(A) I and II

(B) III and IV

(C) IV only

(D) II and IV

Solution by Happy Mittal[edit]

This can be solved using havel hakimi theorem, which says :

  1. First arrange degree sequence in decreasing order.
  2. Remove 1st vertex, and let its degree be k, then subtract 1 from next k vertices.
  3. If all vertices have degree 0, then answer is yes i.e. given degree seqeunce can be a degree sequence for a graph. If any vertex has degree < 0, then answer is no, otherwise repeat step 2.

So we check each degree sequence given in question :

  1. 7, 6, 5, 4, 4, 3, 2, 1. Here first vertex has degree 7, so remove this first vertex, and then subtract 1 from next 7 vertices, so we get 5,4,3,3,2,1,0. Then we get 3,2,2,1,0,0, then 1,1,0,0,0, and then 0,0,0,0. So answer is yes.

Since (I) comes only in one option i.e. option (A), it is correct answer. We don't need to check for other sequences. </p>




blog comments powered by Disqus