---
title: Induced Matching Width in Graph Decompositions
url: https://www.emergentmind.com/topics/induced-matching-width
type: topic
---

# Induced Matching Width in Graph Decompositions

Induced matching width denotes a family of graph decomposition parameters in which structural complexity is measured by the size of induced matchings visible to a decomposition part rather than by the cardinality of that part itself. In recent tree-decomposition work, the precise notion is **induced matching treewidth**, denoted \(\operatorname{tree\textnormal{-}\mu}(G)\), \(\operatorname{imtw}(G)\), or, in the notation of one 2025 paper, \(\yolov(G)\); in the branch-decomposition literature, the corresponding cut-based notion is **maximum induced matching width**, or **mim-width**. These notions are related by theme rather than by definition: induced matching treewidth measures the largest induced matching whose edges all meet one bag of a tree decomposition, whereas mim-width measures the largest induced matching crossing a cut of a branch decomposition [2511.03864] [2402.15834] [1606.08087].

## 1. Formal definitions and terminological scope

All graphs in the tree-decomposition setting discussed here are finite, simple, and undirected. A tree decomposition of a graph \(G\) is a pair \((T,\beta)\) consisting of a tree \(T\) and a map \(\beta:V(T)\to 2^{V(G)}\) such that every edge of \(G\) is contained in some bag \(\beta(t)\), and for every vertex \(v\in V(G)\), the set of nodes \(t\) with \(v\in \beta(t)\) induces a nonempty subtree of \(T\). For a vertex set \(X\subseteq V(G)\), the bag-local induced matching measure is
\[
\mu(G,X)=\max\{|M| : M \text{ is an induced matching in } G \text{ and } e\cap X\neq \emptyset \text{ for all } e\in M\}.
\]
Thus a bag is required to intersect every matching edge, but it need not contain both endpoints of any such edge. For a tree decomposition \(\mathcal T=(T,\beta)\),
\[
\mu_G(\mathcal T)=\max\{\mu(G,\beta(t)) : t\in V(T)\},
\]
and the induced matching treewidth is
\[
\operatorname{tree\textnormal{-}\mu}(G)=\operatorname{imtw}(G)=\min_{\mathcal T}\mu_G(\mathcal T).
\]
This is the parameter studied explicitly in the recent graph-theoretic line initiated from Yolov’s hypergraph width notion [2511.03864] [2402.15834].

The branch-decomposition notion is different. If \((T,\mathcal L)\) is a branch decomposition of \(G\), every edge of \(T\) induces a cut \((A,V(G)\setminus A)\), and the corresponding cut value is the maximum size of an induced matching in the bipartite graph \(G[A,V(G)\setminus A]\). The mim-width of the decomposition is the maximum such cut value, and the mim-width of \(G\) is the minimum over all branch decompositions. The linear variant, LMIM-width, restricts the decomposition to a linear layout [1710.07148] [1509.03753].

| Parameter | Decomposition type | Local measure |
|---|---|---|
| \(\operatorname{tree\textnormal{-}\mu}(G)\), \(\operatorname{imtw}(G)\) | Tree decomposition | Largest induced matching whose every edge meets one bag |
| \(mimw(G)\) | Branch decomposition | Largest induced matching in a crossing bipartite graph |
| \(lmimw(G)\) | Linear layout | Same cut measure, restricted to prefix/suffix cuts |

A persistent source of confusion is the phrase “induced matching width” itself. In the tree-decomposition papers, the precise target notion is induced matching treewidth; in the branch-decomposition papers, “induced matching width” usually refers to mim-width or one of its variants. The shared motif is induced-matchability, but the decomposition frameworks, proof techniques, and algorithmic consequences are distinct.

## 2. Comparison with tree-independence number and related parameters

The closest tree-decomposition analogue of induced matching treewidth is the **tree-independence number**. For a tree decomposition \(\mathcal T=(T,\beta)\),
\[
\alpha_G(\mathcal T)=\max\{\alpha(G[\beta(t)]): t\in V(T)\},
\]
and
\[
\operatorname{tree}\alpha(G)=\min_{\mathcal T}\alpha_G(\mathcal T).
\]
The inequality
\[
\operatorname{imtw}(G)\le \operatorname{tree}\alpha(G)
\]
is immediate from the definitions: if every edge of an induced matching meets a bag \(X\), then selecting one endpoint in \(X\) from each edge yields an independent set in \(G[X]\). The converse fails badly. For the balanced biclique \(K_{n,n}\),
\[
\operatorname{imtw}(K_{n,n})=1 \qquad\text{while}\qquad \operatorname{tree}\alpha(K_{n,n})=n.
\]
Thus induced matching treewidth is strictly more permissive than tree-independence number on dense biclique-like graphs [2402.15834].

Yolov’s identities sharpen this relation by transporting it to derived graphs:
\[
\operatorname{imtw}(G)=\operatorname{tree}\alpha(L^2(G)), \qquad \operatorname{tree}\alpha(G)=\operatorname{imtw}(G\odot K_1).
\]
Here \(L^2(G)\) is the square of the line graph, and \(G\odot K_1\) is the corona obtained by attaching one leaf to each vertex. These identities explain why approximation and hardness information for tree-independence number can transfer to induced matching treewidth, and why tree-independence number embeds into induced matching treewidth via a simple graph transformation [2402.15834].

The relation to ordinary treewidth is asymmetrical. In general,
\[
\operatorname{tree}(G)\le tw(G)+1
\]
in the notation of the 2025 meta-theorem paper, but complete graphs and complete bipartite graphs still have induced matching treewidth at most \(1\), so the parameter is not a sparsity measure in the usual treewidth sense. On bounded-degree graphs, however, the parameters become tightly coupled:
\[
\operatorname{tree}\alpha(G)\le 2\,\operatorname{imtw}(G)\cdot \Delta(G)^2
\]
for every graph with at least one edge, and consequently
\[
\operatorname{tw}(G)\le 2\,\operatorname{imtw}(G)\cdot \Delta(G)^2(\Delta(G)+1).
\]
A plausible implication is that the large gap between induced matching treewidth and more classical sparse-structure parameters is driven primarily by high-degree biclique behavior rather than by tree-decomposition phenomena alone [2402.15834].

## 3. Polynomial equivalence on induced-\(K_{t,t}\)-free classes

A central structural question is when bounded induced matching treewidth forces bounded tree-independence number. Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak had shown that for every two positive integers \(\mu\) and \(t\), there exists an integer \(K(\mu,t)\) such that every \(K_{t,t}\)-free graph \(G\) with \(\operatorname{tree\textnormal{-}\mu}(G)\le \mu\) satisfies \(\operatorname{tree}\alpha(G)<K(\mu,t)\), where \(K_{t,t}\)-free means exclusion as an induced subgraph. The quantitative weakness of that result was that the proof used multiple applications of Ramsey’s theorem, so \(K(\mu,t)\) was exponential in \(\mu\) even for fixed \(t\) [2511.03864].

The 2025 note “Induced matching treewidth and tree-independence number, revisited” replaces those Ramsey-type steps by extremal arguments based on the Kővári--Sós--Turán theorem and proves the polynomial estimate
\[
\operatorname{tree}\alpha(G)=O_t(\mu^{3t^2+1})
\]
for every induced-\(K_{t,t}\)-free graph \(G\) with \(\operatorname{tree\textnormal{-}\mu}(G)\le \mu\). Since \(\operatorname{tree\textnormal{-}\mu}(G)\le \operatorname{tree}\alpha(G)\) always holds, this yields the polynomial equivalence
\[
\operatorname{tree\textnormal{-}\mu}(G)\le \operatorname{tree}\alpha(G)\le O_t(\operatorname{tree\textnormal{-}\mu}(G)^{3t^2+1})
\]
on every class excluding a fixed \(K_{t,t}\) as an induced subgraph [2511.03864].

The proof keeps the overall decomposition strategy of the earlier argument but replaces two extraction lemmas. One turns a large matching in a bipartite graph into either an induced \(K_{t,t}\) or an induced matching of size \(s+1\), with threshold
\[
M(s,t)=O_t(s^t).
\]
The other turns many large independent sets into one global independent set intersecting each of them substantially, with threshold
\[
N(s,t,m)=O_t((sm^2)^t).
\]
Both substitutions are polynomial for fixed \(t\), and both derive from Kővári--Sós--Turán plus a Turán-type sparse-graph independent-set bound.

The same paper also establishes a limit on what kind of polynomial dependence is possible. For any positive integer \(t\), there exists a graph \(G\) that is \(K_{t,t}\)-free and \(tK_2\)-free, but
\[
\operatorname{tree}\alpha(G)=2^{\Omega(t)}.
\]
Since \(tK_2\)-free implies \(\operatorname{tree\textnormal{-}\mu}(G)<t\), no bound polynomial in both \(\operatorname{tree\textnormal{-}\mu}(G)\) and \(t\) can exist in general. The open question left by the paper asks whether for every fixed \(\mu\) there exists a polynomial \(\mathsf p_\mu\) such that every \(K_{t,t}\)-free graph \(G\) with \(\operatorname{tree\textnormal{-}\mu}(G)\le \mu\) satisfies
\[
\operatorname{tree}\alpha(G)\le \mathsf p_\mu(t).
\]
This suggests that the correct asymptotic regime is “fixed \(t\), polynomial in \(\mu\)” rather than a symmetric polynomial dependence on both parameters [2511.03864].

## 4. Algorithmic consequences of bounded induced matching treewidth

The first algorithmic result for induced matching treewidth is due to Yolov: for fixed \(k\), Maximum-Weight Independent Set is solvable in time \(n^{\mathcal O(k)}\) on graphs \(G\) with \(\operatorname{imtw}(G)\le k\). The structural reason is that if a tree decomposition \(\mathcal T\) satisfies \(\mu(\mathcal T)\le k\), then for each bag \(X_t\) one can enumerate in time \(n^{\mathcal O(k)}\) a family \(I_t\) of at most \(n^{2k+2}\) subsets of \(X_t\) such that for every maximal independent set \(I\) of \(G\),
\[
I\cap X_t\in I_t.
\]
Bounded induced matching treewidth therefore yields a polynomially bounded catalogue of relevant bag intersections even when bags are large [2402.15834].

This idea was extended substantially in “Tree decompositions meet induced matchings: beyond Max Weight Independent Set”. For fixed \(k\), Max Weight Induced Forest, equivalently Min Weight Feedback Vertex Set, is solvable on \(n\)-vertex graphs with \(\operatorname{imtw}(G)\le k\) in time \(n^{\mathcal O(k)}\). The same paper proves polynomial-time solvability of Max Weight Independent Packing on bounded-imtw classes, a PTAS for the largest induced subgraph of bounded treewidth, and polynomial-time solvability of even-distance packing via closure under odd powers. It also formulates a CMSO\(_2\)-style conjecture for bounded induced matching treewidth and proves the corresponding theorem on the subclass of graphs of bounded tree-independence number. On the structural side, induced matching treewidth is shown to be monotone under induced minors, and the class \(\operatorname{imtw}(G)\le 1\) is characterized by
\[
\operatorname{imtw}(G)\le 1 \quad\Longleftrightarrow\quad L^2(G)\text{ is chordal}.
\]
On the negative side, deciding whether \(\operatorname{imtw}(G)\le k\) is NP-complete for every fixed \(k\ge 4\), and the paper establishes strong inapproximability results [2402.15834].

The 2025 paper “Finding sparse induced subgraphs on graphs of bounded induced matching treewidth” resolves the meta-algorithmic conjecture in full. It proves that Maximum-Weight Induced Subgraph of Bounded Treewidth is polynomial-time solvable when \(\mathsf{tree\textnormal{-}\mu}(G)\), the target treewidth bound \(w\), and the CMSO\(_2\)-sentence size are bounded. The running time on \(n\)-vertex graphs with \(\mathsf{tree\textnormal{-}\mu}(G)\le k\) is
\[
f(k,w,|\Phi|)\cdot n^{O(kw^2)}.
\]
This strictly subsumes several earlier cases: \(w=0\) yields Maximum-Weight Independent Set, \(w=1\) yields Maximum-Weight Induced Forest, and further choices of \(w\) and \(\Phi\) capture problems such as Weighted Longest Induced Cycle. The technical core is an “inner tree decomposition” of the unknown solution, aligned with the given decomposition of \(G\), whose width is only \(O(kw^2)\) [2507.07975].

## 5. Branch-decomposition variants: mim-width, linear mim-width, sim-width, and one-sided mim-width

The branch-decomposition line studies induced matchings across cuts rather than around bags. Given a branch decomposition \((T,\mathcal L)\), the cut value at an edge \(e\in E(T)\) is the size of a maximum induced matching in the crossing bipartite graph \(G[A_e,\overline{A_e}]\); the maximum over decomposition edges is the decomposition’s mim-width, and minimization over decompositions gives \(mimw(G)\). This framework supports polynomial-time dynamic programming once a bounded-width decomposition is provided. In particular, Longest Induced Path, Induced Disjoint Paths, and fixed-\(H\) Induced Topological Minor admit \(n^{O(w)}\)-time algorithms on graphs of mim-width at most \(w\), given a decomposition, and the same \(n^{O(w)}\) regime holds for Weighted Feedback Vertex Set [1708.04536] [1710.07148].

The linear restriction, LMIM-width, is the cut measure induced by a single vertex order. It is already nontrivial on trees: “Linear MIM-Width of Trees” gives an \(O(n\log n)\) algorithm computing the linear maximum induced matching width of a tree and an optimal layout. The same line also supports output-sensitive enumeration: on graphs of bounded LMIM-width, all 1-minimal and all 1-maximal \((\sigma,\rho)\)-dominating sets can be enumerated with polynomial delay, and unit square graphs have locally bounded LMIM-width with
\[
\operatorname{lmimw}(G[N_G^r[u]])=O(r^2),
\]
which yields an incremental polynomial-time algorithm for enumerating minimal dominating sets [1907.04132] [1509.03753].

A broader family of induced-matching-based cut parameters includes **sim-width** and **one-sided mim-width**. Sim-width strengthens the notion of “induced across the cut” by requiring inducedness in the whole graph rather than only in the crossing bipartite graph; one-sided mim-width, defined by
\[
\omim(A)=\min(\umim(A),\umim(\overline A)),
\]
satisfies
\[
\sw(A)\le \omim(A)\le \mim(A)
\]
for every cut. Chordal graphs and co-comparability graphs have sim-width at most \(1\), but their mim-width is unbounded in general; bounded mim-width reappears on \((K_t\boxminus S_t)\)-free chordal graphs and \((K_t\boxminus K_t)\)-free co-comparability graphs. One-sided mim-width generalizes both bounded mim-width and bounded tree-independence number, and given a branch decomposition of o-mim-width \(k\), Independent Set and Feedback Vertex Set are solvable in time \(n^{\mathcal O(k)}\) [1606.08087] [2302.10643].

## 6. Structural limits, classification results, and current directions

The decomposition-given nature of these parameters is a recurrent limitation. For induced matching treewidth, if \(\operatorname{imtw}(G)\le k\), then one can compute in time \(n^{\mathcal O(k)}\) a tree decomposition \(\mathcal T\) with
\[
\mu(\mathcal T)\le 8k,
\]
but exact recognition is hard: for every fixed \(k\ge 4\), deciding whether \(\operatorname{imtw}(G)\le k\) is NP-complete, and the parameter admits strong hardness of approximation. The same paper also records that bounded induced matching treewidth is incomparable with several other frameworks: complete graphs and complete bipartite graphs can have value \(1\), yet restricted classes such as \(P_4\)-free graphs may still have unbounded induced matching treewidth [2402.15834].

On the mim-width side, hereditary classification has become a central topic. “On algorithmic applications of sim-width and mim-width of \((H_1,H_2)\)-free graphs” gives a complete dichotomy for the classes
\[
(rP_1,\overline{K_{s,t}+P_1})\text{-free},
\]
showing that bounded mim-width occurs exactly when \(r=3\) and one of \(s,t\) is at most \(3\), or \(r=4\) and one of \(s,t\) is at most \(2\). The same paper gives substantial partial classifications for
\[
(K_r,sP_1+tP_2+uP_3)\text{-free}
\]
graphs, and shows that bounded, quickly computable sim-width implies polynomial-time solvability of List \(k\)-Colouring. It also establishes that if Independent Set is polynomial-time solvable on a class of bounded, quickly computable sim-width, then Independent \(\mathcal H\)-Packing is polynomial-time solvable there as well; this includes Induced Matching as the special case \(\mathcal H=\{P_2\}\) [2205.15160].

A common misconception is that all induced-matching-based width notions are interchangeable. The literature instead points to a layered picture. Tree-decomposition parameters such as induced matching treewidth control bag-visible induced matchings and lead to tree-automata and CMSO-style meta-theorems. Branch-decomposition parameters such as mim-width control cut interactions and have generated a different dynamic-programming toolkit. Sim-width and one-sided mim-width enlarge the modeled classes but weaken direct algorithmic leverage; this is precisely why decomposition computation remains a central unresolved issue. In one response to that obstacle, neighbor-depth was introduced as a decomposition-free parameter, and any \(n\)-vertex graph of sim-width \(k\) has neighbor-depth \(\mathcal O(k\log^2 n)\), implying a quasipolynomial-time algorithm for Independent Set on bounded sim-width, and therefore on bounded o-mim-width, even without a supplied decomposition [2302.10643].

The present state of the subject therefore has two focal points. One is **structural calibration**: understanding when induced-matching-based parameters collapse to tree-independence number, treewidth, or bounded-state decompositions under hereditary exclusions such as induced bicliques. The other is **algorithmic extraction**: converting the combinatorial fact “only small induced matchings are visible here” into enumerability of bag or cut signatures, and then into polynomial-time algorithms for induced-subgraph, packing, domination, or logical optimization problems.

Source: https://www.emergentmind.com/topics/induced-matching-width