Explicit minimal embedded resolutions of divisors on models of the projective line

Let $K$ be a discretely valued field with ring of integers ${\mathcal{O}}_K$ with perfect residue field. Let $K(x)$ be the rational function field in one variable. Let ${\mathbb{P}}_{{\mathcal{O}}_K}^1$ be the standard smooth model of ${\mathbb{P}}_K^1$ with coordinate $x$ on irreducible special fiber. Let $f(x) \in $\mathcal{O}$_K[x]$ be a monic irreducible polynomial with corresponding divisor of zeroes ${\text{div}}_0(f)$ on ${\mathbb{P}}_{{\mathcal{O}}_K}^1$. We give an explicit description of the minimal embedded resolution $\mathcal{Y}$ of the pair $({\mathbb{P}}_{{\mathcal{O}}_K}^1,{\text{div}}_0(f))$ by using Mac Lane’s theory to write down the discrete valuations on $K(x)$ corresponding to the irreducible components of the special fiber of $\mathcal{Y}$.