Outermost boundaries for star-connected components in percolation

Tile $\mathbb{R}^2$ into disjoint unit squares $\{S_k\}_{k \geq 0}$ with the origin being the centre of $S_0$ and say that $S_i$ and $S_j$ are star-adjacent if they share a corner and plus-adjacent if they share an edge. Every square is either vacant or occupied. If the occupied plus-connected component $C^+(0)$ containing the origin is finite, it is known that the outermost boundary $\partial^+_0$ of $C^+(0)$ is a unique cycle surrounding the origin. For the finite occupied star-connected component $C(0)$ containing the origin, we prove in this paper that the outermost boundary $\partial_0$ is a unique connected graph consisting of a union of cycles $\cup_{1 \leq i \leq n} C_i$ with mutually disjoint interiors. Moreover, we have that each pair of cycles in $\partial_0$ share at most one vertex in common and we provide an inductive procedure to obtain a circuit containing all the edges of $\cup_{1 \leq i \leq n} C_i.$ This has applications for contour analysis of star-connected components in percolation.