Solving Partial Dominating Set and Related Problems Using Twin-Width

Abstract

Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are $\rm W[1]$-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form $\phi\equiv\exists x_1\cdots \exists x_k \sum_{\alpha \in I} \#y\,\psi_\alpha(x_1,\ldots,x_k,y)\ge t$, where $\psi_\alpha$ is a quantifier-free formula for each $\alpha \in I$, $t$ is an arbitrary number, and $\#y$ is a counting quantifier, can be evaluated in time $f(d,k)n$, where $n$ is the number of vertices and $d$ is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.