On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions.

💡 Research Summary

The paper addresses the Maximum Cardinality Cut (Max‑Cut) problem, which asks for a partition of a graph’s vertex set that maximizes the number of edges crossing the cut. While Max‑Cut is NP‑hard in general, it is polynomial‑time solvable for several special graph families (planar, line graphs, bounded clique‑width, etc.). However, two previously published polynomial‑time algorithms claiming to solve Max‑Cut on proper interval graphs were later shown to be flawed, leaving the complexity status for this class open.

The authors introduce two new graph parameters that stem from a generalized “bubble” representation, and they develop fixed‑parameter tractable (FPT) algorithms parameterized by these measures. A bubble is defined as a maximal set of twin vertices (vertices sharing the same neighbourhood up to each other). Contracting each bubble to a single vertex yields a graph G⁻ whose vertices correspond to bubbles; the contraction of a bubble partition results in a tree structure T.

For a bubble partition V = {V₁,…,V_k}, they define:

• Independence number α(V) – the maximum size of an independent set inside any part V_i (i.e., the largest number of mutually non‑adjacent bubbles within a part).
• Width w(V) – the maximum number of bubbles contained in any part V_i.

The α‑bubble width bw_α(G) is the smallest w(V) over all partitions with α(V) ≤ α.

A key structural result (Theorem 2) shows that every graph possesses a tight maximum cut, meaning that the set of bubbles crossed by an optimal cut forms an independent set in G⁻. This property enables a dynamic‑programming algorithm that processes the tree T bottom‑up. For each node t, the algorithm enumerates all possible configurations γ ∈ Γ(V_t) describing how the cut interacts with the bubbles of V_t (which bubbles are crossed, how many vertices of each crossed bubble lie on each side, and which bubbles are entirely on one side). The number of configurations is bounded by |V_t|·2^{α(V_t)}·2^{|V_t⁻|}, leading to an overall running time

  O*(|V(G)|⁴·α(V)⁴·w(V))

as stated in Theorem 3. This yields an FPT algorithm when α and w are treated as parameters.

The second contribution is the introduction of an (α,β,δ)-clique‑width decomposition. Traditional clique‑width measures the minimum number of labels needed to construct a graph using four operations (vertex creation, disjoint union, edge insertion between label classes, and label renaming). The (α,β,δ) variant augments this by bounding:

• α – the independence number of each label class (as above),
• β – the maximum number of bubbles per label, and
• δ – a restriction on how bubbles of adjacent labels may intersect.

The authors prove (Theorem 5) that the XP algorithm for Max‑Cut on bounded clique‑width graphs (previously known) becomes FPT when parameterized by the smallest width cw_{α,β,δ}(G) of an (α,β,δ)-decomposition. Moreover, any bubble partition with independence number α can be transformed into an (α,1,1)-clique‑width decomposition of comparable width, showing that the result of Section 3 is a special case of the more general framework of Section 4.

The paper further explores the relationship between the class BW₁ (graphs admitting a bubble partition with α = 1) and several well‑studied families: proper interval graphs, mixed unit‑interval graphs, chordal graphs, co‑bipartite graphs, and split graphs. For mixed unit‑interval graphs, a bubble partition with α = 1 can be found in polynomial time, implying that the presented FPT algorithm applies directly to this class.

In the concluding section, the authors list open problems, such as: (i) efficiently computing a bubble partition that simultaneously minimizes α and w, (ii) establishing tight lower and upper bounds for cw_{α,β,δ}(G) on broader graph families, and (iii) extending the bubble‑based approach to other graph classes (e.g., triangle‑free or bounded‑dimensional intersection graphs).

Overall, the paper corrects earlier misconceptions about the tractability of Max‑Cut on proper interval graphs, introduces a novel structural tool (bubble partitions) together with a refined clique‑width notion, and provides the first FPT algorithms for Max‑Cut parameterized by these new measures. The results not only settle the status for proper and mixed unit‑interval graphs but also open a pathway for parameterized algorithms on a wide spectrum of graph classes where traditional width parameters are insufficient.