<p>You can make it valid depending on your definition of a valid network flow.
Let s and t be the source and sink, respectively. Depending on your problem you may no further restrictions of the network flow conservation of t and s. In most definitions there are even no further restrictions on that, as this would just complicate the definition (without helping much).</p>

<p>If you want to restrict such "superficial" network flows in your example, you may however further state that for each edge (u,v) with a flow (i.e. flow function f(u,v)>0) there must be a path from s to u where each edge on the path has a positive flow (i.e. f(e)>0).</p>

<p>You may further restrict the flow on s not to be negative and the flow on t not beeing positive, i.e. for sum for all u on f(s,u) must be positive and sum for all u on f(u,t) must be positive or 0.</p>

<p>This should then remodel a "more intuitive" network flow comparable with water flows :).</p>

<p>(Note that the restriction: the source must not send out any transport units does not help much to prevent such superficial flows as suggested in the question.)</p>

<p>However, typically one assumes that f(t,u)&lt;=0 for all vertices u, i.e. the sink consumes everything which it retrieves.</p>

CC BY-SA 3.0
2013-03-01T17:43:35.477Z
2013-03-01T17:43:35.477Z
eci
<p>I have a network flow (with min and max capacities) where the only transport units flowing are within a cycle (of flow value 2). The source of the network does not send out any transport units into any direction and the transport units getting into the sink are all getting sent away directly, so 0 transport units "stay" there. Now my question is, is such a flow allowed as long as minimium neccesarry flow (given by min capacities) exists and the amount of incoming transport units is equal to the amount of outgoing transport units? Or is a flow network where the source doesn´t send out any transport units not permitted? </p>

<p>I look forward for any help.</p>

CC BY-SA 3.0
2013-03-02T07:12:03.007Z
Welcome to cstheory, a Q&A site for *research-level* questions in *theoretical computer science* (TCS). Your question does not appear to be a research-level question in TCS. Please see the [FAQ] for more information on what is meant by this. Your question might be suitable for [cs.se] which has a broader scope.
Kaveh
CC BY-SA 3.0
2013-03-02T12:20:23.307Z
I am sorry, I did not know this. Thanks for clarification. I will consider reposting this question in the Computer Science section.
Philipp
CC BY-SA 3.0
2013-03-02T21:13:03.143Z
If you want I can migrate the question to [cs.se] for you.
Kaveh
CC BY-SA 3.0
2013-03-03T09:37:49.007Z
Thank you Kaveh, since this question and the question related to it (see http://cstheory.stackexchange.com/questions/16664/using-max-flow-ford-fulkerson-to-find-satisfying-flow) have already been answered it is not necessary to migrate it. However, since it might help other people struggling with similar problems, it might be a good idea to migrate both questions to the CS section.
Philipp
<graph-theory><graph-algorithms>
Source sending 0 transport units
Philipp
# Source sending 0 transport units

(Asked by: username_0 on 2013-03-01)

I have a network flow (with min and max capacities) where the only transport units flowing are within a cycle (of flow value 2). The source of the network does not send out any transport units into any direction and the transport units getting into the sink are all getting sent away directly, so 0 transport units "stay" there. Now my question is, is such a flow allowed as long as minimium neccesarry flow (given by min capacities) exists and the amount of incoming transport units is equal to the amount of outgoing transport units? Or is a flow network where the source doesn´t send out any transport units not permitted?

I look forward for any help.

**Tags:** graph-theory, graph-algorithms
> **Comments:**
> **username_1**: Welcome to cstheory, a Q&A site for *research-level* questions in *theoretical computer science* (TCS). Your question does not appear to be a research-level question in TCS. Please see the [FAQ] for more information on what is meant by this. Your question might be suitable for [cs.se] which has a broader scope.  
> **username_0**: I am sorry, I did not know this. Thanks for clarification. I will consider reposting this question in the Computer Science section.  
> **username_1**: If you want I can migrate the question to [cs.se] for you.  
> **username_0**: Thank you Kaveh, since this question and the question related to it (see http://cstheory.stackexchange.com/questions/16664/using-max-flow-ford-fulkerson-to-find-satisfying-flow) have already been answered it is not necessary to migrate it. However, since it might help other people struggling with similar problems, it might be a good idea to migrate both questions to the CS section.  

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## 1 answers

### Answer by username_2 on 2013-03-01. Score: 1 | Accepted answer

You can make it valid depending on your definition of a valid network flow.
Let s and t be the source and sink, respectively. Depending on your problem you may no further restrictions of the network flow conservation of t and s. In most definitions there are even no further restrictions on that, as this would just complicate the definition (without helping much).

If you want to restrict such "superficial" network flows in your example, you may however further state that for each edge (u,v) with a flow (i.e. flow function f(u,v)>0) there must be a path from s to u where each edge on the path has a positive flow (i.e. f(e)>0).

You may further restrict the flow on s not to be negative and the flow on t not beeing positive, i.e. for sum for all u on f(s,u) must be positive and sum for all u on f(u,t) must be positive or 0.

This should then remodel a "more intuitive" network flow comparable with water flows :).

(Note that the restriction: the source must not send out any transport units does not help much to prevent such superficial flows as suggested in the question.)

However, typically one assumes that f(t,u)<=0 for all vertices u, i.e. the sink consumes everything which it retrieves.


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