Arjun Suresh (talk | contribs) |
Arjun Suresh (talk | contribs) |
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be the degree sequence of any graph? | be the degree sequence of any graph? | ||
| − | I. 7, 6, 5, 4, 4, 3, 2, 1 | + | I. $7, 6, 5, 4, 4, 3, 2, 1$ |
| − | II. 6, 6, 6, 6, 3, 3, 2, 2 | + | II. $6, 6, 6, 6, 3, 3, 2, 2$ |
| − | III. 7, 6, 6, 4, 4, 3, 2, 2 | + | III. $7, 6, 6, 4, 4, 3, 2, 2$ |
| − | IV. 8, 7, 7, 6, 4, 2, 1, 1 | + | IV. $8, 7, 7, 6, 4, 2, 1, 1$ |
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<ol> | <ol> | ||
<li>First arrange degree sequence in decreasing order.</li> | <li>First arrange degree sequence in decreasing order.</li> | ||
| − | <li>Remove | + | <li>Remove $1^{st}$ vertex, and let its degree be $k$, then subtract $1$ from next $k$ vertices.</li> |
| − | <li>If all vertices have degree 0, then answer is yes i.e. given degree | + | <li>If all vertices have degree $0$, then answer is yes i.e. given degree sequence can be a degree sequence for a graph. |
| − | If any vertex has degree < 0, then answer is no, otherwise repeat | + | If any vertex has degree < $0$, then answer is no, otherwise repeat |
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 | + | 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> |
</ol> | </ol> | ||
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. | ||
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.
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.
GATECSE