Graphs are often used as models to solve problems in computer science, mathematics, and biology. A pancake sorting problem is modeled using a pancake graph whose classes include burnt pancake graphs, signed permutation graphs, and restricted pancake graphs. The network cost is degree × diameter. Finding a graph with a small network cost is like finding a good sorting algorithm. We propose a novel recursively divided pancake (RDP) graph that has a smaller network cost than other pancake-like graphs. In the pancake graph P_n, the number of nodes is n!, the degree is n − 1, and the network cost is O(n²). In an RDP_n, the number of nodes is n!, the degree is 2log₂n − 1, and the network cost is O(n(log₂n)³). Because O(n(log₂n)³) < O(n²), the RDP is superior to other pancake-like graphs. In this paper, we propose an RDP_n and analyze its basic topological properties. Second, we show that the RDP_n is recursive and symmetric. Third, a sorting algorithm is proposed, and the degree and diameter are derived. Finally, the network cost is compared between the RDP graph and other classes of pancake graphs.

中文翻译：

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具有较小网络成本的递归划分的煎饼图

图形通常用作解决计算机科学，数学和生物学问题的模型。煎饼分类问题是使用煎饼图来建模的，该煎饼图的类包括烧饼图，有符号排列图和受限煎饼图。网络成本为度×直径。查找具有较低网络成本的图就像查找良好的排序算法。我们提出了一种新颖的递归分割煎饼（RDP）图，该图的网络成本比其他类似煎饼的图要小。在煎饼图P _n中，节点数为n !，度为n -1，网络成本为O（n ²）。在RDP _n中，节点数为n！，度为2log ₂ n -1，网络成本为O（n（log ₂ n）³）。因为O（n（log ₂ n）³）< O（n ²），所以RDP优于其他煎饼状图。在本文中，我们提出了一个RDP _n并分析了其基本拓扑特性。其次，我们证明RDP _n是递归和对称的。第三，提出了一种排序算法，并推导了度数和直径。最后，在RDP图和其他类别的煎饼图之间比较网络成本。