کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
437499 | 690150 | 2011 | 12 صفحه PDF | دانلود رایگان |
The Inscribed Square Conjecture has been open since 1911. It states that any plane Jordan curve J contains four points on a non-degenerate square. In this article two different discrete versions of this conjecture are introduced and proved. The first version is in the field of digital topology: it is proved that the conjecture holds for digital simple closed 4-curves, and that it is false for 8-curves. The second one is in the topological graph theory field: it is proved that any cycle of the grid Z2 contains an inscribed square with integer vertices. The proofs are based on a theorem due to Pak. An infinite family of 4-curves in the digital plane containing a single non-degenerate inscribed square is introduced as well as a second infinite family containing one 4-curve with exactly n inscribed squares for each positive integer value of n. Finally an algorithm with time complexity O(n2) is given to find inscribed squares in simple digital curves.
Journal: Theoretical Computer Science - Volume 412, Issue 15, 25 March 2011, Pages 1301-1312