Induced Minor-Free Graphs

• Induced Minor-Free Graphs are defined as graphs that exclude a fixed graph H as an induced minor, meaning no sequence of vertex deletions and edge contractions can yield H.
• They display significant structural dichotomies, including bounded clique-width and well-quasi-ordering properties, which aid in solving problems like graph isomorphism and Maximum Independent Set.
• These graphs support universal constructions, balanced separator theorems, and explicit extremal bounds that underpin efficient algorithms and sparse regime analyses.

An induced minor-free graph is one that excludes a fixed graph $H$ as an induced minor, meaning no sequence of vertex deletions and edge contractions (without creating new adjacencies) yields $H$ as a subgraph. This relation is stricter than the ordinary graph minor relation and is central to a range of structural, algorithmic, and universality phenomena in modern graph theory.

1. Formalism and Key Variants

Let $H$ and $G$ be finite graphs. $H$ is an induced minor of $G$ if there exist pairwise-disjoint connected subgraphs $\{X_v \subseteq V(G): v\in V(H)\}$ such that for $u,v\in V(H)$, $X_u$ and $X_v$ are adjacent in $G$ if and only if $uv\in E(H)$; equivalently, $H$ can be obtained from $G$ via vertex deletions and edge contractions, with no new edges between otherwise nonadjacent branch-sets (Belmonte et al., 2016, Bousquet et al., 2024).

Graph classes are said to be $H$-induced-minor-free if no such model exists. $$\mathcal{C}_H = \{G : H\text{ is not an induced minor of } G\}$$ is hereditary and closed under both vertex deletion and edge contraction. Notably, the induced minor relation resides strictly between induced subgraph and minor inclusion.

2. Structural Dichotomies and Decomposition Theory

Induced minor-free classes are structurally rich and admit certain dichotomies. The survey by Belmonte, Otachi, and Schweitzer provides two complete classifications:

• Graph Isomorphism Dichotomy: GI is polynomial for $H$-induced-minor-free graphs if and only if $H$ is a complete graph, an induced subgraph of $\mathrm{co}$-$P_3\cup 2K_1$, or the gem; otherwise, GI-complete (Belmonte et al., 2016).
• Clique-width Dichotomy: $H$-induced-minor-free classes have bounded clique-width exactly when $H$ is an induced subgraph of the gem or $\mathrm{co}$-$P_3\cup 2K_1$ (Belmonte et al., 2016).

Further, well-quasi-ordering is characterized: the class of $H$-induced-minor-free graphs is wqo by induced minor if and only if $H$ is an induced minor of either the gem or $K_4^+$ (the complete graph on 4 vertices plus a vertex of degree 2) (Błasiok et al., 2015).

Induced minor decompositions mirror canonical minor decompositions, but exclude certain connectors (e.g., arbitrary edge deletions) and demand new techniques. For instance, in $K_4$-minor-free graphs, blocks decompose into wheels, cycles, or multipartite graphs, and for gem-free graphs, a bounded number of cographs and paths remain after deletion of a small set (Błasiok et al., 2015).

3. Universality and Infinite Constructions

One profound structural result is the existence of universal graphs for classes avoiding certain minors. For every $k \ge 3$, there exists a countable $W_k$-minor-free graph $U_k$ that is universal in the induced subgraph sense: every smaller $W_k$-minor-free graph embeds as an induced subgraph in $U_k$ (Krill, 2023). The construction exploits 2-edge-colored universal graph “bricks” and a stepwise gluing procedure, followed by edge deletions to enforce forbidden minors without sacrificing universality.

Analogous universality results hold for $C_n$-minor-free graphs and select other excluded minors. These systems connect induced minor theory to universal algebraic and model-theoretic methods.

4. Sparse Regimes: Treewidth, Tree-independence, and Separator Theorems

Induced minor-free classes can admit dense graphs; nevertheless, exclusion in conjunction with bounded degree or star-free constraints recovers strong sparsity properties.

• Separator Theorem: For any fixed $H$, every $H$-induced-minor-free $n$-vertex $m$-edge graph has a balanced vertex separator of order $O_H(\sqrt{m})$, with constructive randomized algorithms for finding either a minor model of $H$ or such a separator (Korhonen et al., 2023).
• Bipartite and Pathwidth Barriers: In weakly sparse classes (e.g., $K_{t,t}$-subgraph-free), the excluded induced forest minor theorem classifies forests $H$ for which all weakly sparse $H$-induced-minor-free graphs have bounded pathwidth. Only induced minors from two explicit infinite families of parameterized trees qualify (Bonnet et al., 1 Dec 2025).
• Tree-independence Number: Via bramble-duality, for $K_{1,d}$-free graphs excluding a planar induced minor (notably wheels and ladders), the tree-independence number $\alpha$-tw is bounded by an explicit function of $d$ and the forbidden minor’s parameters. This leads to polynomial time algorithms for classical $\mathsf{NP}$-hard problems within these classes, including Maximum Independent Set (Choi et al., 10 Jun 2025, Choi et al., 4 Sep 2025).

For graphs with bounded maximum degree and large treewidth, the existence of large induced grid minors holds, generalizing the classical grid minor theorem (Korhonen, 2022). Consequently, for fixed planar $H$, subexponential algorithms for Maximum Weight Independent Set on $H$-induced-minor-free graphs are available.

5. Explicit Extremal and Balance Properties

Extremal results further quantify induced minor-free subgraph sizes:

• For connected $m$-edge graphs, the largest induced $K_4$-minor-free subgraph has size at least $n - \frac{3}{16}(m + 1)$ (sharp), equivalently bounding treewidth at most 2 (Joret et al., 2016).
• For general $K_h$-minor-free induced subgraphs, tightness results imply size bounds of $n-m/6+o(m)$. For planar and partial 2-tree induced minors similar constants are obtained (Borradaile et al., 2014).

Balanced separator theorems with domination properties have been established for wheel-induced-minor-free graphs: every such graph admits either a separator of bounded size or a separator dominated by a bounded number of vertices, confirming the Gartland–Lokshtanov conjecture for wheels (Chudnovsky et al., 13 Dec 2025).

6. Algorithmic and Quasi-isometric Implications

Structural results directly inform algorithmic applications:

• Polynomial and subexponential-time algorithms for detecting forbidden induced minors in bounded $\alpha$-tw classes (Korhonen et al., 2023, Choi et al., 10 Jun 2025, Bousquet et al., 2024).
• For $K_{2,3}$-induced-minor-free graphs, an explicit quasi-isometry to graphs of treewidth at most 2 with additive distortion $c$ exists; layered tree-based embeddings and bounded strong isometric path complexity assure metric approximation (Chakraborty, 2 Mar 2025).
• Dichotomy results ground the tractability of the graph isomorphism problem in structural forbidden minors (Belmonte et al., 2016).

Recognition of forbidden induced minors is generally $\mathsf{NP}$-complete for certain fixed graphs, and no algorithm of complexity $2^{o(n/\log^3 n)}$ for generic induced minor testing exists under ETH (Korhonen et al., 2023).

7. Open Problems and Future Directions

• Tightening Treewidth and Separator Bounds: Closing the gap between subpolynomial treewidth bounds (currently $2^{c\log^{1-\varepsilon} n}$) and the conjectured polylogarithmic bounds for induced-minor-free classes is a major direction (Chudnovsky et al., 21 Dec 2025).
• Representability and Region Intersection Graphs: Not all induced-minor-free classes are region intersection graphs over minor-free classes, even for small excluded minors; new combinatorial representations may be needed (Bonnet et al., 29 Apr 2025).
• Extending Dichotomies: Fully classifying pairs (and larger sets) of excluded induced minors governing wqo, clique-width, and GI complexity remains open.
• Full Induced Grid Theorem: Extending wheel and ladder results to structural induced grid minors in $K_{1,d}$-free graphs via $\alpha$-treewidth is conjectured but incomplete (Choi et al., 4 Sep 2025).

Induced minor-free graph theory interleaves combinatorial, algorithmic, and metric properties, driving research across universality, quasi-ordering, and sparse structure theory. These interactions point toward a rich landscape of undetermined structural and computational phenomena.