**Hinged dissections of polyominos and polyforms**.

E. Demaine, M. Demaine, D. Eppstein, G. Frederickson, and E. Friedman.

arXiv:cs.CG/9907018.

*11th Canad. Conf. Comp. Geom.,*1999.

*Computational Geometry: Theory and Applications*31 (3): 237–262, 2005 (special issue for 11th CCCG).We show that, for any n, there exists a mechanism formed by connecting polygons with hinges that can be folded into all possible n-ominos. Similar results hold as well for n-iamonds, n-hexes, and n-abolos.

(BibTeX – Erik's CCCG publication page – Erik's CGTA publication page – Citations)

**Phutball endgames are hard**.

E. Demaine, M. Demaine, and D. Eppstein.

arXiv:cs.CC/0008025.

*More Games of No Chance*, R. J. Nowakowski, ed., MSRI Publications 42, pp. 351–360.We show that, in John Conway's board game Phutball (or Philosopher's Football), it is NP-complete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time.

(BibTeX – Citations – Erik's publications page – CiteSeer)

**Flat foldings of plane graphs with prescribed angles and edge lengths**.

Z. Abel, E. Demaine, M. Demaine, D. Eppstein, A. Lubiw, and R. Uehara.

arXiv:1408.6771.

*22nd Int. Symp. Graph Drawing*, Würzburg, Germany, 2014.

Springer,*Lecture Notes in Comp. Sci.*8871, 2014, pp. 272–283.Given a plane graph with fixed edge lengths, and an assignment of the angles 0, 180, and 360 to the angles between adjacent edges, we show how to test whether the angle assignment can be realized by an embedding of the graph as a flat folding on a line. As a consequence, we can determine whether two-dimensional cell complexes with one vertex can be flattened. The main idea behind the result is to show that each face of the graph can be folded independently of the other faces.

**Folding polyominoes into (poly)cubes**.

O. Aichholzer, M. Biro, E. Demaine, M. Demaine, D. Eppstein, S. P. Fekete, A. Hesterberg, I. Kostitsyna, and C. Schmidt.

*27th Canadian Conference on Computational Geometry*, Kingston, Ontario, Canada, 2015, pp. 101–106.

arXiv:1712.09317.

*Int. J. Comp. Geom. & Appl.*, to appear.We classify the polyominoes that can be folded to form the surface of a cube or polycube, in multiple different folding models that incorporate the type of fold (mountain or valley), the location of a fold (edges of the polycube only, or elsewhere such as along diagonals), and whether the folded polyomino is allowed to pass through the interior of the polycube or must stay on its surface.

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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