We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for $K_2$-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which $K_2$-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite $K_2$-hypohamiltonian graphs. Furthermore, we describe a new operation for creating $K_2$-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar $K_2$-hypohamiltonian graph of order $n$ for every integer $n\geq 134$. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence $K_2$-hypohamiltonian graphs with large maximum degree and size.

翻译：本文提出一种算法，能够生成给定阶数下所有两两不同构的$K_2$-次哈密顿图，即移除任意一对相邻顶点后得到哈密顿图的非哈密顿图。我们引入了专门针对$K_2$-次哈密顿图设计的全新界定准则，从而改进了先前的计算结果。具体而言，我们刻画了存在$K_2$-次哈密顿图的阶数特征，并提升了最小平面$K_2$-次哈密顿图与最小二分$K_2$-次哈密顿图阶数的现有下界。此外，我们描述了一种创建$K_2$-次哈密顿图的新操作，该操作在特定条件下可保持平面性，并利用它证明了对于每个整数$n\geq 134$，均存在阶数为$n$的平面$K_2$-次哈密顿图。最后，受Thomassen关于次哈密顿图定理的启发，我们证明了存在具有大最大度与大尺寸的$K_2$-次哈密顿图。