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I'm not fully understanding the concept of retracts, I think. For example, does this graph have any possible retracts? It seems like it doesn't to me, but I'm not sure how to check.

Also, what if instead of each of $2, 4, 6, 8, 10,$ and $12,$ there were paths of arbitrarily long length between $1$ and $3, 3$ and $5, 5$ and $7, 7$ and $9, 9$ and $11,$ and $11$ and $1$? Would there be retracts then?

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Angelo
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casi
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  • Also, what if instead of 2, 4, 6, 8, 10, and 12 we had long paths of arbitrary length? I'm just trying to get an idea of how retracts work and look. – casi Aug 24 '22 at 20:54
  • In the usual topic called "graph theory" the length of edges is not part of the definition. Sometimes the edges are given an orientation, from one vertex to the other, but that is called a "directed graph" and has other definitions and theorems. The number of vertices is positive and finite. [There are some cases where infinite vertex sets are considered but then some term other than "graph" is used. I'm not familiar with those] for a "simple graph" there are not "parallel edges" [two or more edges joining the same two vertices]. There are also no loops. – coffeemath Aug 25 '22 at 00:37
  • Please note my example before (now deleted) is not correct. A retract of graph G onto a subgraph H must map the set of all vertices of G into the vertex set of H, and preserve edges. [My example before did not define the subgraph, and also did not map all vertices of G to vertices of H. In the wiki article on this they note that the map of vertices of G to vertices of H does not need to be injective. They have a nice example displayed prominently near the beginning. https://en.wikipedia.org/wiki/Graph_homomorphism – coffeemath Aug 25 '22 at 01:11

1 Answers1

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It's just a hint.

For example, the induced subgraph $H=\{1,2,5\}$ is a retract. The corresponding retraction is $\{2,4,6,8,10,12\}\to2$, $\{1,3,7\}\to1$, $\{5,9,11\}\to5$.

kabenyuk
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