XNLP-completeness for Parameterized Problems on Graphs with a Linear Structure

In this paper, we show several parameterized problems to be complete for the class XNLP: parameterized problems that can be solved with a non-deterministic algorithm that uses $f(k)\log n$ space and $f(k)n^c$ time, with $f$ a computable function, $n$ the input size, $k$ the parameter and $c$ a constant. The problems include Maximum Regular Induced Subgraph and Max Cut parameterized by linear clique-width, Capacitated (Red-Blue) Dominating Set parameterized by pathwidth, Odd Cycle Transversal parameterized by a new parameter we call logarithmic linear clique-width (defined as $k/\log n$ for an $n$-vertex graph of linear clique-width $k$), and Bipartite Bandwidth.