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I am new to this forum, and I am hoping that there is a mathematical solution to this situation. I am trying to form a network structure of parent and children nodes, but with some conditions. Following is a description:

Assume a finite number of nodes (Let's say 50 or take any finite one number. In an application of this structure, the real number might be very high). Now a closed structure of these nodes is to be constructed such that

  1. Every node always has a single parent (there can be multiple parents in the network)
  2. Every node always has a minimum of one child, there is no limit to the maximum number of children. In short, every node is a parent to atleast one node. If not every node, then a maximum number of nodes have atleast one child.

Here is an example of a decentralized network. But it is an open network with end nodes without children, whereas I am looking at possibly a closed network with minimum or zero "childless" nodes.

Can such a structure exist? Any thoughts would be really appreciated.

Thanks!

user3606960
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  • Take a triangle with vertices $a,b,c$. Say $a$ is a parent of $b$ which is a parent of $c$ which is a parent of $a$. Does that work for your definition of "parent" and "closed"? – Arkady May 10 '16 at 08:09
  • I'm not sure that even the notions of "parent" and "child" are compatible with the closed structure. In binary trees, for instance, these terms make sense after specifying a choice of root, because there is only one path to each vertex from the root. If the graph is not a tree, there may be loops which prevent this interpretation. [EDIT: such as the situation Jake describes.] – Eric Stucky May 10 '16 at 08:10
  • Welcome to mathSE. Could you insert details into your question so that it's a perfectly well-defined mathematical question? I'm worried that you left out the definitions of several key concepts and didn't try to frame your question in a fully mathematical way. – Caleb Stanford May 10 '16 at 08:11
  • Jake, it definitely works for a small number of nodes, like you described for 3 vertices, yes. But for a larger number of nodes, where there is a possibility to have more than one children for every node, this approach doesn't seem to make justice. – user3606960 May 10 '16 at 08:24
  • Eric, the definition of parent and child are definitely visually easy to interpret in binary trees, but assuming that I can have any node as my child, except for siblings and immediate parent, the structure can have loops and still follow the parent & child notions. – user3606960 May 10 '16 at 08:28
  • 6005, I understand and apologise that my question is not framed in a fully mathematical way. But I am afraid that I am not a mathematics person, I come from the creative field, and hence unaware of the correct mathematical terms for this. But I hope I could explain further if there is anything specific that I couldn't make clear in my description? – user3606960 May 10 '16 at 08:35

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