1. NP-Completeness of the Combinatorial Distance Matrix Realisation Problem
- Author
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Fairbairn, David L., Mertzios, George B., and Peyerimhoff, Norbert
- Subjects
Computer Science - Data Structures and Algorithms ,Computer Science - Computational Complexity ,Computer Science - Discrete Mathematics ,F.2.2 ,G.2.2 - Abstract
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., Comment: 27 pages, 5 figures more...
- Published
- 2024