# A note on independent sets in trees

TL;DR: A simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by m( T), a result first proved by Wilf.

Abstract: We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \[ m( T ) = \begin{cases} 2^{k - 1} + 1& {\text{if }} n = 2k, \\ 2^k & {\text{if }} n = 2k + 1, \end{cases}\] a result first proved by Wilf [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 125–130]. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.

## Summary (1 min read)

### Summary

- The authors also characterize those trees achieving this maximum value.
- Finally the authors investigate some related problems.
- Key words, independent vertices, trees, extremal graphs AMS(MOS) subject classifications.

Did you find this useful? Give us your feedback

...read more

##### Citations

1,041 citations

62 citations

46 citations

### Additional excerpts

...Sagan [26] finally presented an elegant proof, in which trees attaining the upper bound were also found (as did Griggs and Grinstead [7] independently)....

[...]

46 citations

41 citations

### Cites result from "A note on independent sets in trees..."

...Quite a lot of similar results are given in the graph-theoretic literature as well: for instance, Hedman [4] studies the (essentially equivalent) problem of maximizing the number of cliques in graphs with a given maximal clique size, and Wilf [21] gives the largest number of maximal independent vertex sets of a tree on n vertices, see also [15]....

[...]

##### References

94 citations

80 citations