# A Gluing Scheme for Maximal-Girth Tree-Decomposed Graphs

**Minisymposium:**
COMBINATORICS

**Content:**
If we take three perfect binary trees of the same height we can build a trivalent graph, that we will call \emph{quasi-binary tree}, by adding an edge from each one of the three root vertices of the original trees to a new vertex.
We will show that, for all $n \in \mathbb{N}$, there is a way of gluing three quasi-binary trees of the same height $n$ such that the girth of the connected graph, that results from identification of the leaves, has the maximal possible value, that is: $2n+2$.