The graph \(G'\) obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence \(G_0, G_1, G_2, \ldots , G_k\) of graphs such that \(G_0=G\) and \(G_i\) is a 1-fold of \(G_{i-1}\) for each \(i=1, 2, \ldots , k\) is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is \(n-3\). Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.

中文翻译：

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关于图的折叠厚度

通过识别G中具有至少一个公共相邻点的两个不相邻顶点从图G获得的图\（G'\）称为G的1倍。序列\（G_0，G_1，G_2，\ ldots，G_k \）的曲线图，使得\（G_0 = G \）和\（的G_i \）是一个1倍\（G_ {I-1} \）为如果序列中的图全为奇异或全非奇异，则每个\（i = 1、2，\ ldots，k \）都称为G的均匀k折叠。的折叠厚度ģ是最大ķ为其中有一个均匀的ķ的-折叠摹。我们在这里显示n阶奇异二部图的折叠厚度为\（n-3 \）。此外，非奇异二部图的折叠厚度为0，即，非奇异二部图的每1倍是奇异的。我们还确定了一些著名的图形族的折叠厚度，例如循环，风扇和一些轮子。此外，我们研究了通过对这些图族执行操作而获得的图的折叠厚度。具体来说，我们确定从两个图的笛卡尔积获得的图的折叠厚度以及其成分均为同构的不连续图的折叠厚度。