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In this paper, we investigate a particular variation of the valid order problem, which is derived from the map folding problem. Our focus is on folding a grid pattern augmented with half of the diagonal creases to form a regular grid pattern via simple folds. The conclusion is that, given an overlapping order for all the boundary triangular faces of the grid pattern, it is possible to determine in \(O(m+n)^2\) time whether a simple folding process can achieve a compatible flat-folded state, with the boundary triangular faces overlapping in the given order.

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Validity of Boundary Orders in Flat-Folding 1-Diagonal Grid Patterns

  • 会议论文

  • 发表日期：

    2026年

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  • Yiyang Jia,
  • Jun Mitani

摘要

In this paper, we investigate a particular variation of the valid order problem, which is derived from the map folding problem. Our focus is on folding a grid pattern augmented with half of the diagonal creases to form a regular grid pattern via simple folds. The conclusion is that, given an overlapping order for all the boundary triangular faces of the grid pattern, it is possible to determine in \(O(m+n)^2\) time whether a simple folding process can achieve a compatible flat-folded state, with the boundary triangular faces overlapping in the given order.