Optimal antithickenings of claw-free trigraphs

Chudnovsky and Seymour’s structure theorem for claw-free graphs has led to a multitude of recent results that exploit two structural operations: {\em compositions of strips} and {\em thickenings}. In this paper we consider the latter, proving that every claw-free graph has a unique optimal {\em antithickening}, where our definition of {\em optimal} is chosen carefully to respect the structural foundation of the graph. Furthermore, we give an algorithm to find the optimal antithickening in $O(m^2)$ time. For the sake of both completeness and ease of proof, we prove stronger results in the more general setting of trigraphs.

💡 Research Summary

The paper investigates the inverse operation of “thickening” in the context of claw‑free and quasi‑line graphs, extending the study to the more general framework of trigraphs. A trigraph is defined by a vertex set together with a function θ that assigns each unordered pair of distinct vertices a value of 1 (strong adjacency), 0 (semi‑adjacency) or –1 (strong anti‑adjacency); semi‑adjacent pairs form a matching. Within this setting the authors focus on homogeneous pairs of strong cliques (HPOSC), which are two disjoint strong cliques A and B such that every vertex outside A∪B is either strongly complete or strongly anticomplete to each of A and B. Two refined notions are introduced: deletion‑minimal HPOSCs, which contain a square (C₄) and have no vertex in A (or B) that is completely or anticompletely adjacent to the opposite side; and square‑connected HPOSCs, which guarantee that any non‑trivial partition of A (or B) is intersected by a square. The paper proves a series of structural lemmas describing how such pairs intersect, showing that skew intersections force the union of the four parts to be covered by two strong cliques and to form a homogeneous set or the whole vertex set. Consequently, a connected claw‑free trigraph containing two skew‑intersecting deletion‑minimal HPOSCs must be degenerate (α≤2 or cobipartite).

The central result (Theorem 1) states that any non‑degenerate claw‑free trigraph (or quasi‑line trigraph that is not cobipartite) possesses a unique optimal antithickening, and that this antithickening can be computed in O(m²) time, where m counts unordered adjacent pairs. An antithickening replaces each semi‑edge by a single edge, effectively collapsing each HPOSC into a single edge. “Optimal” means that the resulting trigraph is as small as possible while preserving the structural foundation required by the Chudnovsky‑Seymour decomposition (i.e., the remainder is a circular‑interval graph or a composition of linear‑interval strips).

To achieve the algorithmic claim the authors adapt the King‑Reed procedure for constructing a square‑connected HPOSC. Starting from a pair of strongly adjacent vertices that belong to a square, they iteratively enlarge the two sides A and B by adding any vertex that is not uniformly (strongly) complete or anticomplete to the current side. The process stops when no such vertex exists, yielding a maximal square‑connected HPOSC that is also deletion‑minimal. By repeatedly extracting maximal HPOSCs and contracting each into a single edge, while ensuring that already contracted semi‑edges are not revisited, the algorithm builds the unique optimal antithickening. Each extraction examines at most O(m) vertices, and there are at most O(m) extractions, giving the O(m²) bound.

The paper also discusses how this result strengthens earlier work by Fáñez, Oriolo, and Snels, who provided a multi‑step reduction for graphs without addressing uniqueness or optimality. By working directly with trigraphs, the authors obtain a one‑step reduction that is both theoretically clean and practically faster. Moreover, the uniqueness of the optimal antithickening implies that any algorithmic application relying on the removal of homogeneous pairs (e.g., coloring, maximum weight independent set, minimum dominating set) can be performed deterministically without ambiguity.

In summary, the authors provide a comprehensive structural and algorithmic treatment of antithickening in claw‑free and quasi‑line trigraphs, proving the existence and uniqueness of an optimal reduction and delivering an O(m²) algorithm to compute it. This advances both the theoretical understanding of claw‑free graph decomposition and the practical toolkit for designing efficient algorithms on these graph classes.