Linear clique-width and modular decomposition

Abstract: A hereditary class of graphs has bounded clique-width if and only if its prime members do, but this lifting property fails for linear clique-width. We prove that a hereditary class has bounded linear clique-width if and only if its prime members do and it contains neither all quasi-threshold graphs nor all complements of quasi-threshold graphs. This generalizes a result of Brignall, Korpelainen, and Vatter, who established the result for cographs.

• The paper establishes that a hereditary graph class has bounded linear clique-width if and only if its prime members have bounded linear clique-width and it excludes both quasi-threshold and co-quasi-threshold graphs.
• The authors introduce universal quasi-threshold graphs to constructively demonstrate unbounded linear clique-width and avoid well-quasi-ordering arguments.
• The findings reduce the verification of bounded linear clique-width in a hereditary class to checking its prime members plus explicit forbidden-subgraph conditions.

Background and motivation

Clique-width, introduced by Courcelle, Engelfriet, and Rozenberg, admits a clean structural lifting property: a hereditary class of graphs has bounded clique-width if and only if its prime members do. This follows directly from the definition, since any graph is an inflation of its prime skeleton and inflations cost at most one extra label per module. Linear clique-width — the sequential variant in which disjoint unions of previously constructed labeled graphs are disallowed — fails to enjoy this property. The cographs ($P_4$-free graphs) are the canonical counterexample: their only prime members are $K_2$ and $\overline{K}_2$, with linear clique-width $2$, yet the class as a whole has unbounded linear clique-width. Brignall, Korpelainen, and Vatter resolved this failure within cographs, showing that the quasi-threshold graphs (trivially perfect graphs) and their complements are the sole obstructions.

The paper under review, by Brignall, Opler, and Vatter (2602.22089), proves that this dichotomy holds for arbitrary hereditary classes, not merely cographs:

Main theorem. A hereditary class of graphs has bounded linear clique-width if and only if its prime members have bounded linear clique-width and it contains neither all quasi-threshold graphs nor all complements of quasi-threshold graphs.

The necessity direction is immediate: quasi-threshold graphs have unbounded linear clique-width (a fact already observed by Brignall, Korpelainen, and Vatter), so any class containing all of them, or all of their complements, has unbounded linear clique-width regardless of its prime members. The substance lies in sufficiency: once a class excludes some quasi-threshold graph and some co-quasi-threshold graph, boundedness on primes lifts to the whole class. Practically, this means that verifying bounded linear clique-width for a hereditary class reduces entirely to checking its prime members plus two forbidden-subgraph conditions.

This result also complements the literature on minimal hereditary classes of unbounded clique-width (Lozin; Collins et al.; Brignall and Cocks; Atminas et al.), many of which are simultaneously minimal for linear clique-width. For such classes, the theorem localizes the source of unboundedness to the prime induced subgraphs.

Universal quasi-threshold graphs

A key technical device is an explicit family of universal graphs. Define $Q_1 = K_1$ and, recursively,

$Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$

where $\ast$ denotes join and $\uplus$ disjoint union; the complements $\overline{Q}_s$ satisfy the dual recursion $$\overline{Q}_s = (K_1 \uplus \overline{Q}_{s-1}) \ast \overline{Q}_{s-1}$$. A straightforward induction shows that every quasi-threshold graph embeds as an induced subgraph of some $K_2$0 (dually for co-quasi-threshold graphs). Since linear clique-width is inherited downward by induced subgraphs and is unbounded on the quasi-threshold graphs, the families $K_2$1 and $K_2$2 themselves have unbounded linear clique-width. These graphs play the role that "inflation-witness" structures played in the earlier cograph proof, but here they are used constructively rather than via well-quasi-ordering.

Proof architecture

The proof avoids well-quasi-ordering entirely — a deliberate departure from the blueprint of Albert, Atkinson, and Vatter for separable permutations, which the earlier cograph result had followed. Alecu, Kanté, Lozin, and Zamaraev had noted that reliance on well-quasi-ordering limited the applicability of that approach; the present argument requires only modular decomposition and the universal graphs above.

Three elementary propositions about lcw expressions drive the argument:

• Inflation bound: if $K_2$3, then $K_2$4, by reserving a fixed label pool for module constructions.
• Sink-label improvement: when $K_2$5 is complete or anti-complete, one label serves as a sink, giving $K_2$6.
• Reordering: any specified vertex can be inserted first at a cost of one additional label.

Combining these yields the pivotal dichotomy: for any inflation $K_2$7 with $K_2$8, either some module satisfies $K_2$9, or two distinct modules $\overline{K}_2$0 both satisfy $\overline{K}_2$1.

The main proposition then states: if every prime induced subgraph of $\overline{K}_2$2 has linear clique-width at most $\overline{K}_2$3 and $\overline{K}_2$4, then $\overline{K}_2$5 contains $\overline{K}_2$6 or $\overline{K}_2$7 as an induced subgraph. The proof proceeds by induction on $\overline{K}_2$8. Applying the first stage of modular decomposition, either a single module carries the full linear clique-width of $\overline{K}_2$9 (induction applies), or two modules $2$0 each have linear clique-width at least $2$1, hence each contains $2$2 and $2$3 by induction. If the skeleton $2$4 is prime, some vertex has mixed adjacency to $2$5 and $2$6, supplying exactly the join-plus-disjoint-union pattern needed to assemble $2$7 or $2$8. If $2$9 is complete or anti-complete, only one operation type is available at the top level, so the argument descends into the largest module, whose skeleton must be of the opposite type (co-components of a disconnected graph yield a complete or prime skeleton; components of a co-disconnected graph yield an anti-complete or prime skeleton). A second application of the dichotomy inside that module produces the missing operation, completing the construction. The reduction to $Q_1 = K_1$0 ensures $Q_1 = K_1$1 is disconnected and $Q_1 = K_1$2 is connected, which is what forces these subgraphs to sit inside single components or co-components.

The authors note explicitly that the bound $Q_1 = K_1$3 is not claimed optimal, though there is no slack in the induction, and the linear dependence on $Q_1 = K_1$4 is tight since $Q_1 = K_1$5 grows linearly in $Q_1 = K_1$6.

Context and related parameters

Linear clique-width sits among several sequential width parameters: it is a restriction of clique-width, closely related to Wanke's NLC-width and Gurski and Wanke's linear NLC-width, and was introduced by Lozin and Rautenbach. Cographs have clique-width at most $Q_1 = K_1$7 but unbounded linear clique-width, illustrating how severely the sequential constraint can inflate the parameter.

The concluding section places the theorem in a broader pattern: when the atoms of a decomposition are controlled, unbounded complexity must arise from a self-embedding mechanism. The universal graphs $Q_1 = K_1$8 embody this concretely — each contains two copies of $Q_1 = K_1$9, one distinguished by a joined $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$0. Ferguson and Vatter's analogous result for lettericity (obstructions being matchings, co-matchings, and stacked paths) fits the same template, as does the excluded grid theorem for treewidth. In permutation-pattern terms, the paper's theorem extends the graph side of the Albert–Atkinson–Vatter correspondence beyond cographs: cographs correspond to separable permutations, quasi-threshold and co-quasi-threshold graphs correspond to the classes $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$1, $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$2, $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$3, $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$4, and modular decomposition corresponds to substitution decomposition.

Limitations and open questions

Two caveats bear noting. First, the quantitative bound $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$5 is admittedly non-optimal; determining the correct dependence of the lifting threshold on $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$6, $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$7, and $Q_t = (K_1 \ast Q_{t-1}) \uplus Q_{t-1},$8 remains open. Second, the permutation-side analogy raises a specific unresolved question flagged by the authors: whether the enumerative results of Albert, Atkinson, and Vatter for subclasses of the separable permutations extend to inflations of geometric grid classes. The graph-theoretic theorem here removes a technical obstacle (the well-quasi-ordering machinery) but does not itself settle that enumerative extension.

Conclusion

The paper establishes that quasi-threshold graphs and their complements are the only obstructions to lifting bounded linear clique-width from prime members to an entire hereditary class, generalizing the cograph case of Brignall, Korpelainen, and Vatter. The proof is notably more direct than its predecessor, replacing well-quasi-order arguments with modular decomposition and explicit universal graphs, and it delivers an explicit — if not tight — quantitative threshold. The result sharpens the diagnostic toolkit for linear clique-width: excluding one quasi-threshold and one co-quasi-threshold graph reduces the boundedness question wholly to prime induced subgraphs.