A tree with n vertices is called gracefuL if its vertices can be labelled with integers 1, 2, .... n such that the absolute value of the difference of the labels of adjacent vertices are all different.
Which of the following trees are graceful?
| A. | (a) and (b) |
| B. | (b) and (c) |
| C. | (a) and (c) |
| D. | (a), (b) and (c) |
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Option: D Explanation :
Caterpillar tree: In graph theory, a caterpillar is a tree in which all the vertices are within distance 1 of a central path.
Theorem: All caterpillars are graceful.
So, (a), (b) and (c) are graceful.
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| A. | (a)-True; (b)-True; (c)-False |
| B. | (a)-True; (b)-False; (c)-False |
| C. | (a)-False; (b)-False; (c)-False |
| D. | (a)-True; (b)-True; (c)-True |
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Option: A Explanation : Click on Discuss to view users comments. |
| Codes: | ||||
| (a) | (b) | (c) | (d) | |
| (A) | (i) | (ii) | (iii) | (iv) |
| (B) | (ii) | (iii) | (i) | (iv) |
| (C) | (iii) | (ii) | (iv) | (i) |
| (D) | (iv) | (iii) | (ii) | (i) |
| A. |
(i) (ii) (iii) (iv)
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| B. | (ii) (iii) (i) (iv) |
| C. | (iii) (ii) (iv) (i) |
| D. | (iv) (iii) (ii) (i) |
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Option: A Explanation : Click on Discuss to view users comments. |
Consider the compound propositions given below as:
(a) p˅~(p˄q)
(b) (p˄~q)˅~(p˄q)
c) p˄(q˅r)
Which of the above propositions are tautologies?
| A. | (a) and (c) |
| B. | (b) and (c) |
| C. | (a) and (b) |
| D. | (a), (b) and (c) |
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Option: D Explanation : Click on Discuss to view users comments. |
