Pathwidth, Bandwidth, and Completion Problems to Proper Interval Graphs with Small Cliques

We study two related problems motivated by molecular biology. Given a graph $G$ and a constant $k$, does there exist a supergraph $G'$ of $G$ that is a unit interval graph and has clique size at most $k$? Given a graph $G$ and a proper $k$-coloring $c$ of $G$, does there exist a supergraph $G'$ of $G$ that is properly colored by $c$ and is a unit interval graph? We show that those problems are polynomial for fixed $k$. On the other hand, we prove that the first problem is equivalent to deciding if the bandwidth of $G$ is at most $k-1$. Hence, it is NP-hard and $W[t]$-hard for all $t$. We also show that the second problem is $W[1]$-hard. This implies that for fixed $k$, both of the problems are unlikely to have an $O(n^\alpha)$ algorithm, where $\alpha$ is a constant independent of $k$. A central tool in our study is a new graph-theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph.

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