Two discrete versions of the Inscribed Square Conjecture and some related problems
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 Z(2) 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(n(2)) is given to find inscribed squares in simple digital curves. (C) 2010 Elsevier B.V. All rights reserved.
inscribed square conjecture digital topology jordan curve theorem simple closed digital curves 4-connectivity 8-connectivity