The $k$-CombDMR problem is that of determining whether an $n \times n$ distance matrix can be realised by $n$ vertices in some undirected graph with $n + k$ vertices. This problem has a simple solution in the case $k=0$. In this paper we show that this problem is polynomial time solvable for $k=1$ and $k=2$. Moreover, we provide algorithms to construct such graph realisations by solving appropriate 2-SAT instances. In the case where $k \geq 3$, this problem is NP-complete. We show this by a reduction of the $k$-colourability problem to the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time solvable problem of tree realisability for a given distance matrix.

翻译：$k$-CombDMR问题旨在判定一个$n \times n$距离矩阵是否可由某个具有$n + k$个顶点的无向图中的$n$个顶点实现。当$k=0$时，该问题存在简单解法。本文证明，该问题在$k=1$和$k=2$时是多项式时间可解的。此外，我们通过求解相应的2-SAT实例，提供了构建此类图实现的具体算法。当$k \geq 3$时，该问题是NP完全的。我们通过将$k$-可着色问题归约至$k$-CombDMR问题来证明此结论。最后，我们讨论了针对给定距离矩阵的树实现这一更简单的多项式时间可解问题。