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- W2153211990 abstract "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." @default.
- W2153211990 created "2016-06-24" @default.
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- W2153211990 date "2011-03-01" @default.
- W2153211990 modified "2023-09-29" @default.
- W2153211990 title "Two discrete versions of the Inscribed Square Conjecture and some related problems" @default.
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- W2153211990 doi "https://doi.org/10.1016/j.tcs.2010.10.004" @default.
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