Arjun Suresh (talk | contribs) |
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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 | + | 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. $7, 6, 5, 4, 4, 3, 2, 1$ | I. $7, 6, 5, 4, 4, 3, 2, 1$ | ||
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step 2.</li> | step 2.</li> | ||
</ol> | </ol> | ||
− | So we check each degree sequence given in question : | + | So, we check each degree sequence given in question : |
<ol> | <ol> | ||
<li>$7, 6, 5, 4, 4, 3, 2, 1$. Here first vertex has degree $7$, so remove this first vertex, and then subtract $1$ from next $7$ | <li>$7, 6, 5, 4, 4, 3, 2, 1$. 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.</li> | + | 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.</li> |
+ | <li>$6, 6, 6, 6, 3, 3, 2, 2$. Here first vertex has degree $6$, so remove this first vertex, and then subtract $1$ from next $6$ | ||
+ | vertices, so we get $5, 5, 5, 2, 2, 1, 2$. Then we get $4, 4, 1, 1, 0, 2$ then $3, 0, 0, -1, 2$. Since degree of a vertex becomes | ||
+ | negative, this degree sequence is not possible.</li> | ||
</ol> | </ol> | ||
− | Since ( | + | Since '''(II)''' comes only in option <b>(A)</b> and <b>(D)</b>, and since <b>(A)</b> contains '''I''', which is correct degree |
+ | sequence, <b>(D)</b> must be right answer. We don't need to check for other sequences. | ||
+ | |||
{{Template:FBD}} | {{Template:FBD}} | ||
− | + | ||
[[Category: GATE2010]] | [[Category: GATE2010]] | ||
− | [[Category: Graph Theory questions | + | [[Category: Graph Theory questions from GATE]] |
− |
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
This can be solved using havel hakimi theorem, which says :
So, we check each degree sequence given in question :
Since (II) comes only in option (A) and (D), and since (A) contains I, which is correct degree sequence, (D) must be right answer. We don't need to check for other sequences.
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
This can be solved using havel hakimi theorem, which says :
So we check each degree sequence given in question :
Since (I) comes only in one option i.e. option (A), it is correct answer. We don't need to check for other sequences.