NP-Completeness of the Combinatorial Distance Matrix Realisation Problem

The k-CombDMR problem is that of determining whether an n × 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 ≥ 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.