On the $p'$-subgraph of the Young graph

Let $p$ be a prime number. In this article we study the restriction to $\mathfrak{S}_{n-1}$ of irreducible characters of degree coprime to $p$ of $\mathfrak{S}_n$. In particular, we study the combinatorial properties of the subgraph $\mathbb{Y}_{p'}$ of the Young graph $\mathbb{Y}$. This is an extension to odd primes of the work done by Ayyer, Prasad and Spallone for $p=2$.