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 fc, 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^α) algorithm, where α 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.