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：我们提出了一种算法，能够生成给定阶数的所有两两非同构的 $K_2$-hypohamiltonian 图，即非哈密顿图，但其删除任意一对相邻顶点后得到的图均为哈密顿图。我们引入了专门针对 $K_2$-hypohamiltonian 图设计的新的边界标准，从而改进了先前的计算结果。具体而言，我们刻画了存在 $K_2$-hypohamiltonian 图的阶数，并提高了最小平面图和最小二分 $K_2$-hypohamiltonian 图阶数的现有下界。此外，我们描述了一种新的操作来创建 $K_2$-hypohamiltonian 图，该操作在特定条件下能保持平面性，并利用它证明了对于每个整数 $n \geq 134$，存在一个阶数为 $n$ 的平面 $K_2$-hypohamiltonian 图。最后，受 Thomassen 关于 hypohamiltonian 图的一个定理的启发，我们证明了存在具有大最大度和大规模（边数）的 $K_2$-hypohamiltonian 图。