| Title:
|
Sets Expressible as Unions of Staircase $n$-Convex Polygons (English) |
| Author:
|
Breen, Marilyn |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
23-28 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $P_i$, $1 \le i \le k$. (English) |
| Keyword:
|
orthogonal polygons |
| Keyword:
|
staircase $n$-convex polygons |
| MSC:
|
52A35 |
| idZBL:
|
Zbl 1244.52009 |
| idMR:
|
MR2920696 |
| . |
| Date available:
|
2011-12-08T09:44:56Z |
| Last updated:
|
2013-09-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141719 |
| . |
| Reference:
|
[1] Breen, M.: A Helly theorem for intersections of sets starshaped via staircase $n$-paths. Ars Combinatoria 78 (2006), 47–63. Zbl 1157.52303, MR 2194749 |
| Reference:
|
[2] Breen, M.: Intersections and unions of orthogonal polygons starshaped via staircase $n$-paths. Monatsh. Math. 148 (2006), 91–100. Zbl 1134.52007, MR 2235357, 10.1007/s00605-005-0345-9 |
| Reference:
|
[3] Breen, M.: Staircase $k$-kernels for orthogonal polygons. Arch. Math. 63 (1994), 182–190. Zbl 0742.52006, MR 1289301, 10.1007/BF01189893 |
| Reference:
|
[4] Breen, M.: Unions of orthogonally convex or orthogonally starshaped polygons.. Geometriae Dedicata 53 (1994), 49–56. Zbl 0814.52002, MR 1299884, 10.1007/BF01264043 |
| Reference:
|
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| Reference:
|
[6] Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P. M., Wills, J. M., (eds.) Handbook of Convex Geometry, vol. A, North Holland, New York (1993), 389–448. Zbl 0791.52009, MR 1242986 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Lay, S. R.: Convex Sets and Their Applications. John Wiley, New York, 1982. Zbl 0492.52001, MR 0655598 |
| Reference:
|
[10] McKinney, R. L.: On unions of two convex sets. Canad. J. Math 18 (1966), 883–886. Zbl 0173.15305, MR 0202049, 10.4153/CJM-1966-088-7 |
| Reference:
|
[11] Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40 (1990), 19–48. Zbl 0705.68082, MR 1047288, 10.1016/0022-0000(90)90017-F |
| Reference:
|
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| . |
