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Let S be an orthogonal polygon in the plane. Assume that S is starshaped via staircase paths, and let K be any component of Ker S, the staircase kernel of S, where KS. For every x in S\K, define W K(x) = {s: s lies on some staircase path in S from x to a point of K}. There is a minimal (finite) collection W(K) of W K(x) sets whose union is S. Further, each set W K(x) may be associated with a finite family U K(x) of staircase convex subsets, each containing x and K, with ∪{U: U in U K(x)} = W K(x). If W(K) = {W K(x 1), ..., W K(x n)}, then KV K ≡ ∩{U: U in some family U K(x i), 1 ≤ in} ⊆ Ker S. It follows that each set V K is staircase convex and ∪{V k: K a component of Ker S} = Ker S. Finally, if S is simply connected, then Ker S has exactly one component K, each set W K(x i) is staircase convex, 1 ≤ in, and ∩{W k(x i): 1 ≤ i ≤ n} = Ker S.

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