Clique-Width for Hereditary Graph Classes

Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on ${\cal G}$. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the Colouring and Graph Isomorphism problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.

💡 Research Summary

This survey paper provides a comprehensive overview of the state of research on clique‑width within hereditary graph classes—those closed under taking induced subgraphs. It begins by motivating width parameters in algorithmic graph theory, contrasting treewidth (effective for sparse graphs) with clique‑width, which remains bounded even for dense structures such as complete graphs. After establishing basic terminology, the authors formally define clique‑width via four graph operations (vertex creation, label assignment, disjoint union, and edge insertion between label classes) and discuss its relationships to other width measures like NLC‑width, rank‑width, MIM‑width, and Boolean‑width, noting the known linear‑factor inequalities that link them.

The core of the paper (Section 4) systematically catalogs known results on the boundedness or unboundedness of clique‑width for a wide variety of hereditary families. For each family the authors specify the forbidden induced subgraph(s) H (or a pair (H₁,H₂)) that define the class, and then present the exact dichotomy: either the class has bounded clique‑width (often with an explicit bound) or it contains a family of graphs whose clique‑width grows without bound (e.g., grids, large complete bipartite graphs). Key families covered include:

• H‑free graphs, where H is a small graph such as P₄, claw (K₁,₃), C₄, 2K₂, etc.;
• H‑free chordal graphs and H‑free split graphs, with detailed characterizations for each possible H on up to four vertices;
• H‑free bipartite graphs, where boundedness holds only for very restricted H (paths of length ≤ 6, stars, etc.);
• Classes closed under complementation, where the simultaneous exclusion of a graph and its complement yields bounded clique‑width;
• (H₁,H₂)-free graphs, where only a handful of pairs lead to bounded width.

For each bounded case the paper outlines constructive proofs, typically by exhibiting a decomposition scheme (e.g., a bounded‑label expression or a tree‑like structure) that can be turned into a polynomial‑time algorithm for recognizing the class and for building the expression. For unbounded cases, the authors present infinite antichains or families of graphs (often parameterized grids or bicliques) whose clique‑width grows linearly with size, thereby establishing lower bounds.

Section 5 translates these structural insights into algorithmic consequences. Leveraging Courcelle’s MSO₁ meta‑theorem (graphs of bounded clique‑width admit linear‑time algorithms for any property definable in monadic second‑order logic without edge quantification), the authors show that a large suite of problems—Dominating Set, Independent Set, various coloring variants—become tractable on any hereditary class with bounded clique‑width. They then focus on two flagship problems:

1. Graph Coloring: For hereditary classes defined by two forbidden induced subgraphs, bounded clique‑width implies a polynomial‑time coloring algorithm; otherwise the problem remains NP‑complete. The paper lists all such (H₁,H₂) pairs where this dichotomy is known.
2. Graph Isomorphism (GI): While GI is in quasi‑polynomial time in general, the authors discuss classes where GI is GI‑complete (i.e., as hard as the general problem) and show that many hereditary classes with unbounded clique‑width fall into this category. Conversely, bounded‑width classes often admit polynomial‑time isomorphism tests via canonical clique‑width expressions.

Section 6 explores a conjectured deep link between bounded clique‑width and well‑quasi‑ordering (wqo) under the induced‑subgraph relation. The authors note that all known hereditary classes that are wqo are also of bounded clique‑width, and that many operations preserving wqo (disjoint union, complement, substitution) also preserve boundedness of clique‑width. Nevertheless, a few open cases remain where wqo is established but the clique‑width status is unknown, suggesting that proving a general equivalence could unlock new structural and algorithmic results.

The paper concludes with a curated list of open problems: determining the exact clique‑width status for several (H₁,H₂)-free classes, developing fixed‑parameter algorithms for computing clique‑width on restricted families, and establishing—or refuting—the conjectured equivalence between wqo and bounded clique‑width for hereditary classes. Overall, the survey not only consolidates a decade of research but also provides a clear roadmap for future work at the intersection of graph structure theory, parameterized complexity, and algorithm design.