展開一個多面體是指在其表面上進行切割，使其表面可以在平面上形成一個單一相連且不自我重疊的多邊形平面。關於展開的方法上存在著各種不同的切割方式，在本論文中我們主要使用延邊展開以及網格展開，延邊展開即為沿著多面體的邊進行切割來展開其表面，而網格展開為先將各個面細切為更小的網格並延這些邊進行切割。
單層網格多面體是一個高度為一單位且表面由單位正方形所組成的多面體，我們關注在於展開虧格大於零的單層網格多面體，我們對於含三個孔洞之單層網格多面體和含四個孔洞之單層矩形邊界網格多面體各提出了一個延邊展開演算法，此外我們對於含矩形孔洞之單層單調邊界網格多面體、含矩形孔洞之單層網格多面體、含單調孔洞之單層單調邊界網格多面體和含單調孔洞之單層網格多面體各提出了一個網格展開演算法。
我們最後提出了兩個未解的問題：「是否任意含孔洞之單層網格多面體可以被延邊展開?」以及「是否任意不含孔洞之網格多面體可以被延邊展開?」。

An unfolding of a polyhedron is a cutting of the polyhedron's surface so that its surface can be flattened into a single connected flat patch on the plane without any self-overlapping. There exist many unfolding methods with different cuttings. The two main method of this thesis are an edge-unfolding which is a cutting of the polyhedron's surface along its edges and a grid-unfolding which is a cutting of the polyhedron's refined surface along its edges.
A one-layer lattice polyhedron is a polyhedron of height one, whose surface is composed of unit squares as its faces. We consider the unfolding problem on the one-layer lattice polyhedron of genus greater than zero. We propose two edge-unfolding algorithms for one-layer lattice polyhedra with three holes and one-layer lattice polyhedra with rectangular boundary and four holes. Besides, we propose four grid-unfolding algorithms for one-layer lattice polyhedra with monotone boundary and rectangular holes, one-layer lattice polyhedra with rectangular holes, one-layer lattice polyhedra with monotone boundary and holes, and one-layer lattice polyhedra with monotone holes. The basic idea of these algorithms is that we use different paths to connect the flattened patches of each hole of the given polyhedron so that no self-overlapping can occur in the final flattened patch.
We leave the question open whether any of the general one-layer lattice polyhedra with holes can be edge-unfolded and the question whether any of the general lattice polyhedra without holes can be edge-unfolded.

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