A platypus graph is a non-hamiltonian graph for which every vertex-deleted subgraph is traceable. They are closely related to families of graphs satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian graphs. In this paper, we first investigate cubic platypus graphs, covering all orders for which such graphs exist: in the general and polyhedral case as well as for snarks. We then present (not necessarily cubic) platypus graphs of girth up to 16---whereas no hypohamiltonian graphs of girth greater than 7 are known---and study their maximum degree, generalising two theorems of Chartrand, Gould, and Kapoor. Using computational methods, we determine the complete list of all non-isomorphic platypus graphs for various orders and girths. Finally, we address two questions raised by the third author in [J. Graph Theory \textbf{86} (2017) 223--243].

中文翻译：

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鸭嘴兽图的结构和计算结果

鸭嘴兽图是一种非哈密顿图，其中每个删除顶点的子图都是可追踪的。它们与满足有关最长路径和最长循环的有趣条件的图族密切相关，例如亚哈密顿图、叶稳定图和最大非哈密顿图。在本文中，我们首先研究了立方鸭嘴兽图，涵盖了此类图存在的所有阶数：在一般情况和多面体情况下以及对于蛇类。然后，我们呈现（不一定是三次）鸭嘴兽图，周长可达 16 周长——而没有已知周长大于 7 的亚汉密尔顿图——并研究它们的最大程度，概括了 Chartrand、Gould 和 Kapoor 的两个定理。使用计算方法，我们确定了各种顺序和周长的所有非同构鸭嘴兽图的完整列表。最后，我们解决了第三作者在 [J. 图论 \textbf{86} (2017) 223--243]。