Canonical cuts of path powers

Abstract

The MaxCut problem aims to find a bipartition of vertices in a given graph such that the number of edges with one end vertex in each part is maximum among all bipartitions. NP-hardness when restricted to interval graphs has been recently announced. Surprisingly, all previously published attempts at polynomial-time algorithms for unit interval graphs turned out to be wrong, which justifies the search for subclasses where MaxCut can be handled. We introduce canonical cuts whose pattern allows an easy computation of the cut size for the power of paths $P_n^k$. Using canonical cuts, we calculate the structure and the size of maximum cuts for $k\leq 5$ and for $n\leq \frac{2}{3}(2k+2)$. We prove that the known size for a maximum cut for reduced co-bipartite chain graphs can be achieved by a canonical cut. We perform computational experiments on each $P_n^k$ graph with $1\leq k\leq n\leq 43$ and show that most of them allow a canonical cut that is maximum. We display a table with the found cases where there is no canonical cut which is a maximum cut. In these graphs, the difference between the maximum cut and some canonical is at most 3 units. This indicates canonical cuts as a good approach to tackle the maximum cut on $P_n^k$ graphs.