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.

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