Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that $$\text{depth}(S\mathrm{/}J(C_n{)}^{(t)})=n-1-\lfloor \frac{tn}{2t+1}\rfloor$$ for all $t\ge 2$ and $n\ge 3$, where $S=K[x_1,\dots ,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.