| Abstract |
We will discuss two symmetry breaking parameters: distinguishing number and fixing number. Despite being introduced independently, they share meaningful connections. In particular, we show that if a tree is 2-distinguishable with order at least 3, it suffices to fix at most 4/11 of the vertices and if a tree is $d$-distinguishable, $d \geq 3$, it suffices to fix at most $\frac{d-1}{d+1}$ of the vertices. We also characterize the $d$-distinguishable trees with radius $r$, for any $d \geq 2$ and $r \geq 1$.
This is joint work with Calum Buchanan, Peter Dankleman, Isabel Harris, Paul Horn, and Emily Rivett-Carnac. |
