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Tile \(\mathbb{R}^2\) into disjoint unit squares \(\{S_k\}_{k \geq 0}\) withthe origin being the centre of \(S_0\) and say that \(S_i\) and \(S_j\) arestar-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-connectedcomponent \(C^+(0)\) containing the origin is finite, it is known that theoutermost boundary \(\partial^+_0\) of \(C^+(0)\) is a unique cycle surroundingthe 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, wehave that each pair of cycles in \(\partial_0\) share at most one vertex incommon and we provide an inductive procedure to obtain a circuit containing allthe edges of \(\cup_{1 \leq i \leq n} C_i.\) This has applications for contouranalysis of star-connected components in percolation.

Outermost boundaries for star-connected components in percolation