Hereditary Graph Classes

• Hereditary graph classes are families of graphs closed under induced subgraphs and defined by minimal forbidden structures, providing a clear framework for graph analysis.
• They enable effective structural decomposition and algorithm design, with parameters like clique-width and tree-width ensuring computational tractability.
• Their properties bridge combinatorial, extremal, and model-theoretic perspectives, driving research in problem complexity, coloring, and Ramsey phenomena.

A hereditary graph class is a collection of (finite) simple graphs that is closed under taking induced subgraphs: if $G$ is in the class and $H$ is an induced subgraph of $G$, then $H$ is also in the class. Equivalently, every hereditary class can be uniquely characterized by a (possibly infinite) set of minimal forbidden induced subgraphs. These classes are foundational in both extremal and algorithmic graph theory, as they provide a uniform framework for modeling properties and substructures, underpin major algorithmic dichotomies, and arise naturally in graph limit theory and combinatorial model theory.

1. Characterization and Structure of Hereditary Graph Classes

The standard formulation asserts that for any hereditary graph class $\mathcal{C}$, there exists a (unique) set of minimal forbidden induced subgraphs $\mathcal{F}_\mathcal{C}$ such that: $\mathcal{C} = \{G: \text{%%%%6%%%% is %%%%7%%%%-free for all %%%%8%%%%}\}$ where $F$-free means $F$ does not appear as an induced subgraph of $G$ (Schweitzer, 2014, Janson, 2011, Singh et al., 14 Mar 2024, Singh et al., 26 Mar 2025). This covers the full range: for example, bipartite graphs are characterized by forbidding all odd cycles, cographs by excluding $P_4$, split graphs by $\{C_4, C_5, 2P_2\}$, and so on.

Classes defined by finitely many forbidden induced subgraphs are called finitely-defined, and the parameters

$$c = \max_{F \in \mathcal{F}_\mathcal{C}} |V(F)|,\;\; k = \max_{F \in \mathcal{F}_\mathcal{C}} |E(F)|$$

provide size bounds for obstructions (Singh et al., 14 Mar 2024). Hereditary classes are algorithmically and structurally well-behaved, enabling finite representations, effective recognition, and tractable combinatorial descriptions.

2. Width and Complexity Measures: Clique-Width, Tree-Width, Mim-Width

Hereditary classes interact richly with major graph width parameters:

• Clique-width: A hereditary class has bounded clique-width if and only if its set of forbidden induced subgraphs is contained in $P_4$ (the 4-vertex path); otherwise, boundedness is classified for all pairs $(H_1,H_2)$ (Dabrowski et al., 2019). Notably, minimal hereditary classes of unbounded clique-width may be finitely-defined and correspond to infinite labelled antichains (Atminas et al., 2015).
• Tree-width: Classical results show that bounded tree-width requires forbidding (as induced subgraphs) all complete graphs, complete bipartite graphs, tripods, and their line graphs. In hereditary classes, infinite antichain phenomena preclude the existence of minimal unbounded tree-width subclasses; e.g., path-star classes defined by infinite words exhibit unbounded tree-width but no minimal hereditary subclass of this property (Cocks, 2023).
• Mim-width: For a single forbidden induced subgraph $H$, the class of $H$-free graphs has bounded mim-width if and only if $H \subseteq_{\rm ind} P_4$; for most $(H_1,H_2)$-free classes, a near-complete classification is established, and there are examples with bounded mim-width but unbounded clique-width (Brettell et al., 2020).

These parameters govern algorithmic tractability: bounded clique-width (resp. mim-width, tree-width) allows many otherwise intractable problems (e.g., Coloring, Graph Isomorphism, MSO-definable) to be solved efficiently on hereditary classes (Dabrowski et al., 2019, Schweitzer, 2014, Brettell et al., 2020).

3. Algorithmic Properties and Problem Complexity

3.1 Isomorphism and Coloring Dichotomies

A central recent result is the dichotomy for Graph Isomorphism on hereditary classes defined by two forbidden induced subgraphs: for all but finitely many pairs $(H_1,H_2)$, isomorphism is either in P or GI-complete (Schweitzer, 2014). The classification is via reduction frameworks (encoding templates), modular decompositions generalized to colored graphs, and new algorithmic criteria such as generalized color valence.

For Coloring and Clique Cover, the complexity on $(H,\overline{H})$-free hereditary classes is classified except for two infinite families ($sP_1 + P_3$ for $s \ge 3$ and $sP_1 + P_4$ for $s \ge 2$); the rest are either in P or NP-complete (Blanché et al., 2016).

3.2 Parameterized and Substructure Problems

The parameterized complexity of "finding a $k$-vertex induced subgraph with hereditary property $\Pi$" is fully classified for major hereditary classes: for many cases, the problem is in P or FPT; for others, it is W[1]-hard, with the main boundary determined by Ramsey-type properties of $\Pi$ and the input class (Eppstein et al., 2021).

In problems such as role coloring and coupon coloring, hereditary classes admit boundary classes (infinite antichains) that precisely mark the transition from tractable to intractable subclasses; every finitely-defined hereditary class outside the "easy" side contains a boundary class as a subclass (Purcell et al., 2018).

3.3 Extremal and Speed Results

The structural template theorems provide factorial speed (i.e., labeled enumeration $\log|X_n| = O(n\log n)$) for classes defined by forbidden star forests and their complements, among others (Atminas, 2017). In general, superfactorial speed implies the presence of arbitrarily large stars, their complements, or clique-unions.

4. Decompositions, Model Theory, and Stability

The theory of hereditary classes now includes tight connections to model theory:

• Decomposition Horizons: Every hereditary class with a quasi-bounded-size decomposition into parts of bounded shrubdepth is a stable class (in the sense of model theory, i.e., no formula orders arbitrarily large half-graphs) (Braunfeld et al., 2022). Conversely, every stable hereditary class of graphs admits such a decomposition.
• Graph Limit Theory: Each hereditary property corresponds to a precise zeroset for induced subgraph densities in limit graphons, yielding measure-theoretic characterizations and connections to random-free properties (Janson, 2011).
• Well-Quasi-Order: Hereditary classes defined by finitely many forbidden induced subgraphs are well-quasi-ordered by the induced subgraph relation only in the bounded clique-width regime (Dabrowski et al., 2019).

5. Relaxations: Edge-Apex, Vertex-Apex, and "Almost-Hereditary" Classes

A significant recent advance concerns the stability of hereditary properties under bounded local edit operations:

• For a hereditary class $\mathcal{G}$ defined by finitely many forbidden induced subgraphs, the classes of graphs one edge addition ($\mathcal{G}^{\rm add}$), one edge deletion ($\mathcal{G}^{\rm epex}$), or one vertex deletion ($\mathcal{G}^{\rm vpex}$) away from $\mathcal{G}$ are all hereditary and finitely-defined, with explicit bounds on the sizes of new forbidden subgraphs (Singh et al., 14 Mar 2024, Singh et al., 26 Mar 2025).
• Moreover, iterating a bounded number of these operations preserves finite forbidden subgraph characterizations, and explicit forbidden subgraph lists are known for classical cases such as split graphs, threshold graphs, and cographs.

The introduction of $(p,q)$-edge split graphs, which admit up to $p$ edge additions in the clique side and $q$ edge deletions in the independent side, further expands the toolkit of finely-controlled hereditary relaxations while retaining finite basis properties.

6. Chromatic and Ramsey-Type Phenomena

Hereditary classes underlie deep questions regarding chromatic bounds:

• The assertion that every $\chi$-bounded hereditary class is poly-$\chi$-bounded (Esperet's conjecture) is false in general, but for hereditary classes defined by forbidden induced subdivisions of the claw (or any tree), polynomial bounds exist on $\chi(G)$ in terms of the largest complete multipartite subgraph $\tau_d(G)$ (Rahimi et al., 9 Dec 2025).
• This is established via recursive applications of Ramsey-type threshold lemmas and structural inductions, yielding explicit recurrence bounds for all such hereditary classes.

In addition, stable hereditary classes satisfy strong Erdős–Hajnal-type properties, such as the existence of large homogeneous sets of size at least $\Omega(|G|^{1/2 - \epsilon})$ for every $\epsilon > 0$ (Braunfeld et al., 2022).

7. Open Problems and Research Directions

Prominent open questions include:

• Whether Graph Isomorphism is polynomial-time solvable for all hereditary classes of bounded clique-width (Schweitzer, 2014).
• Providing a full classification of hereditary classes (especially with two forbidden induced subgraphs) for clique-width, mim-width, and the complexity of core algorithmic problems (Dabrowski et al., 2019, Brettell et al., 2020).
• Characterizing the boundaries of indivisibility among hereditary classes beyond lexicographic-product closures (Guingona et al., 2023).
• Extending the machinery of decomposition horizons and well-quasi-order to other model-theoretic dividing lines (NIP, stability, etc.) via bounded shrubdepth or twin-width decompositions (Braunfeld et al., 2022).

These directions draw upon a deep interplay of combinatorial, algorithmic, extremal, and model-theoretic perspectives, with hereditary graph classes serving as the unifying framework.