Tree-Independence Number in Graph Decompositions

• Tree-independence number is a graph parameter defined via tree decompositions that measures the maximum independent set size within any bag.
• It refines classical treewidth by focusing on independence instead of bag size and is bounded by both the treewidth and the overall independence number.
• Algorithmically, it enables efficient solutions for independent set problems in specialized graph classes and underpins approximation schemes in hereditary and minor-closed graphs.

The tree-independence number is a width-type graph parameter defined via tree decompositions, central to recent developments in structural graph theory and algorithmic graph theory, particularly regarding the optimization of independent set-type problems in special graph classes. It measures the sparsity of the largest independent set appearing in any bag of an optimal tree decomposition. The parameter refines the classical notion of treewidth by focusing not on the size but on the independence number of bags. Tree-independence number closely relates to $(\mathrm{tw},\omega)$-boundedness, and its study has unified, sharpened, and extended the understanding of decompositional width in a wide spectrum of hereditary graph families.

1. Definition and Basic Properties

Let $G=(V,E)$ be a finite, simple graph. A tree decomposition of $G$ is a pair $(T,\beta)$ where $T$ is a tree and $\beta:V(T)\to 2^V$ (assigning bags $\beta(x)$ to tree nodes) such that:

• $\bigcup_{x\in V(T)} \beta(x) = V$.
• For every edge $uv\in E$, there exists $x\in V(T)$ with $\{u,v\}\subseteq \beta(x)$.
• For every $v\in V$, the set $\{x\in V(T): v\in\beta(x)\}$ induces a connected subtree of $T$.

For a bag $\beta(x)$, its (bag-)independence number is $\alpha(G[\beta(x)])$, the size of the largest independent set in $G[\beta(x)]$.

The tree-independence number of $G$, denoted $\mathrm{tree}\text{-}\alpha(G)$ (also written as $\mathrm{tin}(G)$ or $\alpha_t(G)$ in the literature), is defined as

$$\mathrm{tree}\text{-}\alpha(G) = \min_{(T,\beta)\ \text{tree dec. of}\ G} \max_{x\in V(T)} \alpha(G[\beta(x)]).$$

A graph $G$ is chordal if and only if $\mathrm{tree}\text{-}\alpha(G)=1$ (Dallard et al., 2021). In general, $1 \le \mathrm{tree}\text{-}\alpha(G) \le \alpha(G)$, and $$\mathrm{tree}\text{-}\alpha(G) \le \mathrm{tw}(G)+1$$.

2. Relationship to Treewidth, Clique-Number, and Related Parameters

The tree-independence number was introduced independently by Yolov and by Dallard, Milanič, and Štorgel (Dallard et al., 2021, Abrishami et al., 2023). It forms a fundamental link between treewidth and the independence structure of bags. For any graph $G$:

• $$\mathrm{tree}\text{-}\alpha(G) \le \mathrm{tw}(G) + 1$$ since every bag has size at most $\mathrm{tw}(G)+1$.
• $\mathrm{tree}\text{-}\alpha(G) \le \alpha(G)$, since the trivial one-bag decomposition yields the global independence number.
• For every induced minor $H$ of $G$, $$\mathrm{tree}\text{-}\alpha(H) \le \mathrm{tree}\text{-}\alpha(G)$$ (Dallard et al., 2022).
• Complete bipartite graphs $K_{n,n}$ satisfy $\mathrm{tree}\text{-}\alpha(K_{n,n}) = n$, demonstrating that $\mathrm{tree}\text{-}\alpha(G)$ is not, in general, controlled by clique-number.

Product bounds and their limits: While it is always the case that $$\mathrm{tw}(G)+1 \leq \mathrm{tree}\text{-}\alpha(G) \cdot \chi(G)$$ and $$\mathrm{tw}(G)+1 \leq \alpha(G) \cdot \mathrm{tree}\text{-}\chi(G)$$, it was shown that in general $$\mathrm{tw}(G)+1 \leq \mathrm{tree}\text{-}\alpha(G) \cdot \mathrm{tree}\text{-}\chi(G)$$ is false: explicit infinite families of graphs achieve arbitrarily large separation between treewidth and the product of tree-independence and tree-chromatic number (Krause et al., 28 Apr 2025).

Ramsey-theoretic bounds: If $\mathrm{tree}\text{-}\alpha(G)\le k$, then $\mathrm{tw}(G) \le R(\omega(G)+1,\,k+1)-2$, with $R$ denoting the classical Ramsey number (Dallard et al., 2021, Dallard et al., 2022).

3. Structural Results and Class-Specific Behavior

Bounded tree-independence in forbidden subgraph/minor settings: For many classes defined by forbidden substructures, the tree-independence number is bounded, which often strictly generalizes bounded treewidth.

• Induced star exclusion: Excluding $K_{1,s}$ as an induced subgraph (optionally together with certain trees and their line graphs) ensures bounded tree-independence number, and for graphs that are $H$-induced-minor-free and $K_{1,s}$-free, it is at most polylogarithmic in $n$, i.e., $$\mathrm{tree}\text{-}\alpha(G) \le (\log n)^{c_{s,H}}$$ for some $c_{s,H}$ depending on $s,H$ (Chudnovsky et al., 29 Dec 2025).
• Polylogarithmic bounds: Even-hole-free graphs, 3PC-free graphs, wall/line-graph-of-wall-free graphs, and classes excluding thetas/pyramids all admit tree decompositions with bags of independence number at most $\log^{O(1)} n$ (Chudnovsky et al., 2024, Chudnovsky et al., 2024, Chudnovsky et al., 24 Jan 2025, Chudnovsky et al., 18 Sep 2025).
• Line graphs: For $L(K_{m,n})$, the line graph of the complete bipartite, $\mathrm{tree}\text{-}\alpha(L(K_{m,n}))=m$; for $L(K_n)$, $$\mathrm{tree}\text{-}\alpha(L(K_n))= \lfloor n/2 \rfloor$$ (Dallard et al., 2024).

Extremal examples: Complete bipartite graphs serve as extremal examples where treewidth and tree-independence number diverge (Dallard et al., 2021, Dallard et al., 2022).

Hereditary and minor-closed behavior: $\mathrm{tree}\text{-}\alpha$ is hereditary, and minor-closed classes often have $\mathrm{tree}\text{-}\alpha$ controlled by excluded minors (Lokshtanov et al., 22 Jan 2026).

(Treewidth, Clique)-boundedness: For hereditary classes, up to polylogarithmic factors, bounded tree-independence number is essentially equivalent to $(\mathrm{tw},\omega)$-boundedness, i.e., classes where treewidth is polynomially bounded in the clique number (Chudnovsky et al., 16 Oct 2025, Hilaire et al., 19 May 2025).

4. Algorithmic Implications and Decomposition Algorithms

Given a tree decomposition $(T,\beta)$ of bag independence number $k$:

• Maximum Weight Independent Set (MWIS): Can be solved in time $O(n^k \cdot |V(T)|)$ via dynamic programming over the decomposition (Dallard et al., 2021).
• (CMSO$_2$, $\mathrm{tw}$)-parametrized problems: For problems such as MWIS or feedback vertex set restricted to induced subgraphs of treewidth $< t$, dynamic programming on a decomposition of bag independence number $k$ solves these in $n^{O(k)}$ time (Lokshtanov et al., 22 Jan 2026).
• Computation/approximation of $\mathrm{tree}\text{-}\alpha(G)$: An $8$-approximation can be computed in $2^{O(k^2)}n^{O(k)}$ time (Dallard et al., 2022). Exact computation is para-NP-hard for $k\ge 4$, and $n^{O(k)}$-time dependence is essentially tight under Gap-ETH (Dallard et al., 2022).
• Quasipolynomial and subexponential algorithmic regimes: Classes with $\mathrm{tree}\text{-}\alpha(G) = O(\log^c n)$ admit $n^{O(\log^c n)}$-time exact algorithms for MWIS and related problems (Lokshtanov et al., 22 Jan 2026, Chudnovsky et al., 24 Jan 2025, Chudnovsky et al., 16 Oct 2025).

5. Connections with Other Decomposition Parameters

5.1. Tree-Chromatic Number

The tree-chromatic number $\mathrm{tree}\text{-}\chi(G)$ is defined as the minimal $k$ such that $G$ admits a tree decomposition where every bag is $k$-colorable. While $\mathrm{tree}\text{-}\alpha(G)$ and $\mathrm{tree}\text{-}\chi(G)$ are always at most $\mathrm{tw}(G)+1$, their product does not, in general, bound treewidth (Krause et al., 28 Apr 2025).

5.2. Induced Matching Treewidth

Induced matching treewidth is the maximum size of an induced matching such that some bag contains an endpoint of each edge. It is always the case that $$\mathrm{imtw}(G) \le \mathrm{tree}\text{-}\alpha(G)$$. For $K_{t,t}$-free graphs, $\mathrm{tree}\text{-}\alpha(G)$ is polynomially bounded in $\mathrm{imtw}(G)$, with explicit polynomial dependencies following from the Kövári–Sós–Turán theorem (Alon et al., 5 Nov 2025).

5.3. Layered and Fractional Variants

Layered tree-independence number $\ellti(G)$, defined via interleaving a layering with the tree decomposition, is crucial in geometric intersection graphs and separator theory; for example, $\ellti(G)=O(g)$ for $g$-map graphs, $O(1)$ for spherical disk graphs (Dallard et al., 14 Jun 2025). Classes with bounded $\ellti(G)$ admit subexponential or quasi-polynomial-time algorithms for MWIS and related problems.

6. Techniques for Bounding Tree-Independence Number

• Balanced separator and central bag methods: Central to the proofs in recent breakthroughs is the construction of small balanced separators (either in size, independence number, or clique-cover number) via recursive decomposition, often extending the techniques used for bounded treewidth but adapted to the independence structure (Abrishami et al., 2023, Chudnovsky et al., 2024, Chudnovsky et al., 2024, Chudnovsky et al., 24 Jan 2025).
• Domination and strip decompositions: The analysis frequently leverages dominated separators and variants of strip/central-bag or "layered set" technology to control the growth of the parameter under recursion.
• Fractional separator LPs and independence-containers: To handle polylogarithmic bounds and LP-based rounding for separators (e.g., independence-containers as a generalization of maximal cliques), container-type arguments and sampling strategies are used to control all small-independent sets with a manageable family of bounded-independence subsets (Chudnovsky et al., 16 Oct 2025).
• Ramsey- and extremal-type bounds: Many upper bounds and tightness arguments use asymmetric Ramsey's theorem or Kövári–Sós–Turán arguments to relate tree-independence to classical extremal quantities (Dallard et al., 2021, Alon et al., 5 Nov 2025).

7. Open Directions and Broader Implications

• Refinement of bounds: While substantial progress has been made—e.g., polylogarithmic bounds for large structural classes—tightening to $O(\log n)$ or $O(1)$ remains outstanding for many fundamental classes (even-hole-free graphs, wall/line-graph-of-wall-free classes) (Chudnovsky et al., 2024, Chudnovsky et al., 24 Jan 2025).
• Algorithmic optimality: Existing FPT-approximation schemes for $\mathrm{tree}\text{-}\alpha(G)$ are near-optimal under standard complexity assumptions; further improvements would require breakthroughs in parameterized complexity (Dallard et al., 2022).
• From bounded tree-independence to $(\mathrm{tw},\omega)$-boundedness: In many settings (forbidden induced subgraphs, particularly $K_{1,s}$, $K_{t,t}$, and certain trees/line-graphs), bounded tree-independence is equivalent to $(\mathrm{tw},\omega)$-boundedness (Dallard et al., 2024, Hilaire et al., 19 May 2025).
• Covering and fractional fragility: The extension of container and degeneracy-based arguments points toward a unified theory underpinning subexponential and approximation algorithms across broad sparse and geometric graph domains (Dallard et al., 14 Jun 2025).

Table: Key Results for Tree-Independence Number in Select Graph Classes

Graph Class / Exclusion	Bound on $\mathrm{tree}\text{-}\alpha(G)$	Algorithmic Implication
Chordal graphs	$1$	Polynomial MWIS
Even-hole-free	$O(\log^{10} n)$	Quasi-poly MWIS (Chudnovsky et al., 2024)
3PC-free	$O(\log^2 n)$	Quasi-poly MWIS (Chudnovsky et al., 2024)
$S_{t,t,t}, K_{t,t},\mathcal{L}_t$-free	$O(\log^4 n)$	Quasi-poly MWIS (Chudnovsky et al., 24 Jan 2025)
$K_{1,s}$-free + planar $H$-induced-minor	$O((\log n)^{c(s,H)})$	Quasi-poly MWIS (Chudnovsky et al., 29 Dec 2025)
Line graphs $L(K_{n})$	$\lfloor n/2 \rfloor$	Poly for fixed $n$ (Dallard et al., 2024)
Complement of line graphs, $K_{s,s}$-free	$O(1)$	Poly MWIS (Hilaire et al., 19 May 2025)

Research on the tree-independence number has established it as the central invariant capturing when the complexity for independent set-type problems in hereditary classes is fundamentally controlled by the clique structure rather than arbitrary bag size, yielding new decompositional paradigms, algorithmic meta-theorems, and deep connections to the foundations of structural graph theory.