Geodesic
Sets Expressible as Unions of Staircase $n$-Convex Polygons
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 23-28.

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Let $k$ and $n$ be fixed, $k\ge 1$, $n \ge 1$, and let $S$ be a simply connected orthogonal polygon in the plane. For $T \subseteq S, T$ lies in a staircase $n$-convex orthogonal polygon $P$ in $S$ if and only if every two points of $T$ see each other via staircase $n$-paths in $S$. This leads to a characterization for those sets $S$ expressible as a union of $k$ staircase $n$-convex polygons $P_i$, $1 \le i \le k$.
Classification : 52A35
Keywords: orthogonal polygons; staircase $n$-convex polygons 
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Breen, Marilyn. Sets Expressible as Unions of Staircase $n$-Convex Polygons. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 50 (2011) no. 1, pp. 23-28. http://geodesic.mathdoc.fr/item/AUPO_2011__50_1_a2/