On graph classes with logarithmic boolean-width

TL;DR

This paper investigates graph classes with boolean-width logarithmic in the number of vertices, providing polynomial algorithms for problems like MWDS and vertex partitioning on these classes.

Contribution

It identifies several graph classes with boolean-width O(log n) and develops polynomial algorithms leveraging this property.

Findings

Boolean-decompositions of width O(log n) can be found in polynomial time for multiple graph classes.

Circular k-trapezoid graphs have boolean-width O(log n), with explicit decompositions given.

Many graph classes contain graphs with boolean-width Theta(log n).

Abstract

Boolean-width is a recently introduced graph parameter. Many problems are fixed parameter tractable when parametrized by boolean-width, for instance "Minimum Weighted Dominating Set" (MWDS) problem can be solved in $O^*(2^{3k})$ time given a boolean-decomposition of width $k$, hence for all graph classes where a boolean-decomposition of width $O(\log n)$ can be found in polynomial time, MWDS can be solved in polynomial time. We study graph classes having boolean-width $O(\log n)$ and problems solvable in $O^*(2^{O(k)})$, combining these two results to design polynomial algorithms. We show that for trapezoid graphs, circular permutation graphs, convex graphs, Dilworth-$k$ graphs, circular arc graphs and complements of $k$-degenerate graphs, boolean-decompositions of width $O(\log n)$ can be found in polynomial time. We also show that circular $k$-trapezoid graphs have boolean-width $O(\log n)$, and find such a decomposition if a circular $k$-trapezoid intersection model is given. For many of the graph classes we also prove that they contain graphs of boolean-width $\Theta(\log n)$. Further we apply the results from \cite{boolw2} to give a new polynomial time algorithm solving all vertex partitioning problems introduced by Proskurowski and Telle \cite{TP97}. This extends previous results by Kratochv\'il, Manuel and Miller \cite{KMM95} showing that a large subset of the vertex partitioning problems are polynomial solvable on interval graphs.