On graph classes with logarithmic boolean-width

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.

💡 Research Summary

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The paper investigates graph classes whose Boolean‑width—a parameter measuring the logarithm of the number of distinct neighbourhood equivalence classes across cuts—is bounded by O(log n). Boolean‑width, introduced recently, directly controls the size of dynamic‑programming tables used in many FPT algorithms; when a Boolean‑decomposition of width k can be found in polynomial time, any problem solvable in 2^{O(k)}·poly(n) time becomes polynomial for classes where k = O(log n).

The authors first formalise the notion of a “PPT triple” (graph class C, parameter W, problem P) and show that for Boolean‑width, the triple (C, Boolean‑width, Minimum Weighted Dominating Set) is PPT for all classes admitting a polynomial‑time construction of a Boolean‑decomposition of logarithmic width. They then systematically identify several well‑studied graph families that satisfy this condition:

1. Permutation graphs – By sorting vertices according to the upper endpoint of their representing line segments, a caterpillar decomposition is built. For any cut, the set of middle vertices m(A) contains at most one representative for each neighbourhood class, yielding Boolean‑width ≤ log n.

2. Circular permutation graphs – Using a circular intersection model of curves between two concentric circles, a similar ordering yields Boolean‑width ≤ 2·log n.

3. k‑trapezoid graphs – Vertices correspond to k‑trapezoids (k intervals on parallel lines). Ordering by the top interval gives Boolean‑width ≤ k·log n. The authors note that constructing a k‑trapezoid model is NP‑complete for k ≥ 3, but when the model is given the bound holds.

4. Circular k‑trapezoid graphs – Starting from an arbitrary point on the circle, vertices whose trapezoids contain that point are added first, followed by the rest ordered by distance. Any neighbourhood can be represented by at most 2k vertices, giving Boolean‑width ≤ 2k·log n.

5. Convex graphs – These are bipartite graphs admitting an ordering of one side such that neighbours of any vertex on the other side form a consecutive interval. By constructing a total ordering that respects this property, the middle set never contains vertices from the interval side, leading to Boolean‑width ≤ log n.

6. Graphs of bounded Dilworth number – Using the Dilworth decomposition, the authors bound Boolean‑width by the logarithm of the size of the largest clique, which is O(log n) for fixed Dilworth number k.

7. Complements of k‑degenerate graphs – Since a k‑degenerate graph can be eliminated vertex‑by‑vertex with at most k earlier neighbours, its complement has Boolean‑width O(k·log n).

For each class the paper either cites existing linear or polynomial‑time recognition algorithms (e.g., permutation graphs, convex graphs) or discusses the complexity of obtaining the required geometric model. Moreover, the authors prove that these classes indeed contain graphs whose Boolean‑width is Θ(log n) by embedding graphs of rank‑width Ω(√n), establishing that the logarithmic bound is tight up to constant factors.

The second major contribution is the application of the Boolean‑width framework to the family of vertex‑partitioning problems introduced by Proskurowski and Telle (including Independent Set, Dominating Set, Perfect Code, k‑Colouring, H‑Cover, H‑Homomorphism, etc.). Bui‑Xuan, Telle and Vatshelle previously showed that all such problems can be solved in 2^{O(k²)}·poly(n) time given a Boolean‑decomposition of width k. By constructing decompositions where the minimal representative of each neighbourhood class has constant size (as proved for all the graph classes above), the exponent collapses to a constant, yielding truly polynomial‑time algorithms for every vertex‑partitioning problem on these classes.

In summary, the paper unifies a wide range of algorithmic results under the single parameter Boolean‑width. It identifies several natural graph families with logarithmic Boolean‑width, provides constructive polynomial‑time decompositions for them, proves tight lower bounds, and leverages these decompositions to obtain polynomial‑time algorithms for Minimum Weighted Dominating Set and the entire suite of Proskurowski‑Telle vertex‑partitioning problems. This work not only broadens the applicability of Boolean‑width‑based techniques but also suggests that many other problems may become tractable on graph classes where a logarithmic Boolean‑width decomposition can be efficiently constructed.