new graph with adding 1 edges and number

Question

In a preparing class for ACM-Contest, our Teacher has given us one printed page of a solved problem. On one page its written that, 2 of the following facts are True but she won't say, why or what:

"with adding new 1 edge to a Directed Graph, how many of these may be true about number of strongly connected component of this graph?

+) at most 1 unit is increased.

++) at most 1 unit is decreased.

+++) maybe not changed.

++++) maybe decreased by more than 2 units.

anyone could expanded it clearly with our teams?

David Eisenstat · Accepted Answer · 2015-03-17T14:21:27.097

1

(+) is false (under a generous interpretation of "increase at most", i.e., implying that there is no decrease) and (++) is false and (++++) is true because if we add the feedback arc from one end of a path to the other, making it a cycle, then we go from n strong components to 1. (+++) is obviously true.

Before (4 strong components):
* --> * --> * --> *

After version A (1 strong component):
___________________
|                 |
v                 |
* --> * --> * --> *

After version B (4 strong components):
___________________
|                 |
|                 v
* --> * --> * --> *

edited Mar 17 '15 at 14:21

answered Mar 17 '15 at 14:04

David Eisenstat

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I confused about this sentence" because if we add the feedback arc from one end of a path to the other, then we go from n strong components to 1" – Mar 17 '15 at 14:11
@MinoJende We go from an acyclic path to one big cycle. – David Eisenstat Mar 17 '15 at 14:16
is it possible to make an example for this part of your sentence? – Mar 17 '15 at 14:19
wow, really nice, thanks. my last question is about +++? I couldn't understand it ! – Mar 17 '15 at 14:23
@MinoJende Added another diagram. – David Eisenstat Mar 17 '15 at 14:25
you are the best clever person i see – Mar 17 '15 at 14:26

new graph with adding 1 edges and number

1 Answers1