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What is a T-star packing and what is Proper Maximal 3-Star Packing? I read some definitions and can't understand

"By a T-star we mean a complete bipartite graph K1,t for some t ≤ T. For an undirected graph G, a T-star packing is a collection of node-disjoint T-stars in G."

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    $\begingroup$ What in particular is confusing here? $\endgroup$
    – Juho
    Jul 16, 2021 at 7:03

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So a $t$-star is the graph you get by taking one central node and connecting $t$ neighbors to it. Equivalently, a $t$-star is the complete bipartite graph $K_{1,t}$ if you wish to express it like that as your definition does.

A $t$-star packing then is a bunch of such $t$-stars found from some graph $G$ so that their nodes don't overlap. As a simple example, consider the 4-cycle. You can find a 1-star packing of size 2 in it: take one edge as the other 1-star, and the other edge as the other 1-star.

The packing described above is maximal: you can't extend it by adding a new member (i.e., a third 1-star) into it. It also happens to be maximum, i.e., it is the largest such packing you can find. I encourage you to think about what the difference between a maximal and maximum packing is, i.e., find an example where they differ.

I don't what proper here means, but it is likely defined in your source.

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