Trees with proper thinness 2

The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A graph is proper $k$-thin if its vertices can be ordered in such a way that there is a partition of the vertices into $k$ classes satisfying that for each triple of vertices $r < s < t$, such that there is an edge between $r$ and $t$, it is true that if $r$ and $s$ belong to the same class, then there is an edge between $s$ and $t$, and if $s$ and $t$ belong to the same class, then there is an edge between $r$ and $s$. The proper thinness is the smallest value of $k$ such that the graph is proper $k$-thin. In this work we focus on the calculation of proper thinness for trees. We characterize trees of proper thinness~2, both structurally and by their minimal forbidden induced subgraphs. The characterizations obtained lead to a polynomial-time recognition algorithm. We furthermore show why the structural results obtained for trees of proper thinness~2 cannot be straightforwardly generalized to trees of proper thinness~3.