Induced Matching Width in Graph Decompositions

Updated 6 July 2026

Induced matching width is defined as the maximum size of induced matchings intersecting a decomposition bag or cut, capturing local graph structure.
It distinguishes between tree-decomposition (imtw) and branch-decomposition (mim-width) frameworks, each with unique algorithmic and structural implications.
Bounded induced matching width facilitates efficient dynamic programming algorithms for problems like Maximum-Weight Independent Set and related optimization tasks.

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 (Alon et al., 5 Nov 2025, Lima et al., 2024, Kang et al., 2016).

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 $\operatorname{imtw}(G)$ 0 with $\operatorname{imtw}(G)$ 1 induces a nonempty subtree of $\operatorname{imtw}(G)$ 2. For a vertex set $\operatorname{imtw}(G)$ 3, the bag-local induced matching measure is

$\operatorname{imtw}(G)$ 4

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 $\operatorname{imtw}(G)$ 5,

$\operatorname{imtw}(G)$ 6

and the induced matching treewidth is

$\operatorname{imtw}(G)$ 7

This is the parameter studied explicitly in the recent graph-theoretic line initiated from Yolov’s hypergraph width notion (Alon et al., 5 Nov 2025, Lima et al., 2024).

The branch-decomposition notion is different. If $\operatorname{imtw}(G)$ 8 is a branch decomposition of $\operatorname{imtw}(G)$ 9, every edge of $\yolov(G)$0 induces a cut $\yolov(G)$1, and the corresponding cut value is the maximum size of an induced matching in the bipartite graph $\yolov(G)$2. The mim-width of the decomposition is the maximum such cut value, and the mim-width of $\yolov(G)$3 is the minimum over all branch decompositions. The linear variant, LMIM-width, restricts the decomposition to a linear layout (Jaffke et al., 2017, Golovach et al., 2015).

Parameter	Decomposition type	Local measure
$\yolov(G)$4, $\yolov(G)$5	Tree decomposition	Largest induced matching whose every edge meets one bag
$\yolov(G)$6	Branch decomposition	Largest induced matching in a crossing bipartite graph
$\yolov(G)$7	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.

The closest tree-decomposition analogue of induced matching treewidth is the tree-independence number. For a tree decomposition $\yolov(G)$8,

$\yolov(G)$9

and

$G$ 0

The inequality

$G$ 1

is immediate from the definitions: if every edge of an induced matching meets a bag $G$ 2, then selecting one endpoint in $G$ 3 from each edge yields an independent set in $G$ 4. The converse fails badly. For the balanced biclique $G$ 5,

$G$ 6

Thus induced matching treewidth is strictly more permissive than tree-independence number on dense biclique-like graphs (Lima et al., 2024).

Yolov’s identities sharpen this relation by transporting it to derived graphs: $G$ 7 Here $G$ 8 is the square of the line graph, and $G$ 9 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 (Lima et al., 2024).

The relation to ordinary treewidth is asymmetrical. In general,

$(T,\beta)$ 0

in the notation of the 2025 meta-theorem paper, but complete graphs and complete bipartite graphs still have induced matching treewidth at most $(T,\beta)$ 1, so the parameter is not a sparsity measure in the usual treewidth sense. On bounded-degree graphs, however, the parameters become tightly coupled: $(T,\beta)$ 2 for every graph with at least one edge, and consequently

$(T,\beta)$ 3

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 (Lima et al., 2024).

3. Polynomial equivalence on induced- $(T,\beta)$ 4-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 $(T,\beta)$ 5 and $(T,\beta)$ 6, there exists an integer $(T,\beta)$ 7 such that every $(T,\beta)$ 8-free graph $(T,\beta)$ 9 with $T$ 0 satisfies $T$ 1, where $T$ 2-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 $T$ 3 was exponential in $T$ 4 even for fixed $T$ 5 (Alon et al., 5 Nov 2025).

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

$T$ 6

for every induced- $T$ 7-free graph $T$ 8 with $T$ 9. Since $\beta:V(T)\to 2^{V(G)}$ 0 always holds, this yields the polynomial equivalence

$\beta:V(T)\to 2^{V(G)}$ 1

on every class excluding a fixed $\beta:V(T)\to 2^{V(G)}$ 2 as an induced subgraph (Alon et al., 5 Nov 2025).

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 $\beta:V(T)\to 2^{V(G)}$ 3 or an induced matching of size $\beta:V(T)\to 2^{V(G)}$ 4, with threshold

$\beta:V(T)\to 2^{V(G)}$ 5

The other turns many large independent sets into one global independent set intersecting each of them substantially, with threshold

$\beta:V(T)\to 2^{V(G)}$ 6

Both substitutions are polynomial for fixed $\beta:V(T)\to 2^{V(G)}$ 7, 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 $\beta:V(T)\to 2^{V(G)}$ 8, there exists a graph $\beta:V(T)\to 2^{V(G)}$ 9 that is $G$ 0-free and $G$ 1-free, but

$G$ 2

Since $G$ 3-free implies $G$ 4, no bound polynomial in both $G$ 5 and $G$ 6 can exist in general. The open question left by the paper asks whether for every fixed $G$ 7 there exists a polynomial $G$ 8 such that every $G$ 9-free graph $\beta(t)$ 0 with $\beta(t)$ 1 satisfies

$\beta(t)$ 2

This suggests that the correct asymptotic regime is “fixed $\beta(t)$ 3, polynomial in $\beta(t)$ 4” rather than a symmetric polynomial dependence on both parameters (Alon et al., 5 Nov 2025).

4. Algorithmic consequences of bounded induced matching treewidth

The first algorithmic result for induced matching treewidth is due to Yolov: for fixed $\beta(t)$ 5, Maximum-Weight Independent Set is solvable in time $\beta(t)$ 6 on graphs $\beta(t)$ 7 with $\beta(t)$ 8. The structural reason is that if a tree decomposition $\beta(t)$ 9 satisfies $v\in V(G)$ 0, then for each bag $v\in V(G)$ 1 one can enumerate in time $v\in V(G)$ 2 a family $v\in V(G)$ 3 of at most $v\in V(G)$ 4 subsets of $v\in V(G)$ 5 such that for every maximal independent set $v\in V(G)$ 6 of $v\in V(G)$ 7,

$v\in V(G)$ 8

Bounded induced matching treewidth therefore yields a polynomially bounded catalogue of relevant bag intersections even when bags are large (Lima et al., 2024).

This idea was extended substantially in “Tree decompositions meet induced matchings: beyond Max Weight Independent Set”. For fixed $v\in V(G)$ 9, Max Weight Induced Forest, equivalently Min Weight Feedback Vertex Set, is solvable on $\operatorname{imtw}(G)$ 00-vertex graphs with $\operatorname{imtw}(G)$ 01 in time $\operatorname{imtw}(G)$ 02. 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 $\operatorname{imtw}(G)$ 03-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 $\operatorname{imtw}(G)$ 04 is characterized by

$\operatorname{imtw}(G)$ 05

On the negative side, deciding whether $\operatorname{imtw}(G)$ 06 is NP-complete for every fixed $\operatorname{imtw}(G)$ 07, and the paper establishes strong inapproximability results (Lima et al., 2024).

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 $\operatorname{imtw}(G)$ 08, the target treewidth bound $\operatorname{imtw}(G)$ 09, and the CMSO $\operatorname{imtw}(G)$ 10-sentence size are bounded. The running time on $\operatorname{imtw}(G)$ 11-vertex graphs with $\operatorname{imtw}(G)$ 12 is

$\operatorname{imtw}(G)$ 13

This strictly subsumes several earlier cases: $\operatorname{imtw}(G)$ 14 yields Maximum-Weight Independent Set, $\operatorname{imtw}(G)$ 15 yields Maximum-Weight Induced Forest, and further choices of $\operatorname{imtw}(G)$ 16 and $\operatorname{imtw}(G)$ 17 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 $\operatorname{imtw}(G)$ 18, whose width is only $\operatorname{imtw}(G)$ 19 (Bodlaender et al., 10 Jul 2025).

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 $\operatorname{imtw}(G)$ 20, the cut value at an edge $\operatorname{imtw}(G)$ 21 is the size of a maximum induced matching in the crossing bipartite graph $\operatorname{imtw}(G)$ 22; the maximum over decomposition edges is the decomposition’s mim-width, and minimization over decompositions gives $\operatorname{imtw}(G)$ 23. This framework supports polynomial-time dynamic programming once a bounded-width decomposition is provided. In particular, Longest Induced Path, Induced Disjoint Paths, and fixed- $\operatorname{imtw}(G)$ 24 Induced Topological Minor admit $\operatorname{imtw}(G)$ 25-time algorithms on graphs of mim-width at most $\operatorname{imtw}(G)$ 26, given a decomposition, and the same $\operatorname{imtw}(G)$ 27 regime holds for Weighted Feedback Vertex Set (Jaffke et al., 2017, Jaffke et al., 2017).

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 $\operatorname{imtw}(G)$ 28 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 $\operatorname{imtw}(G)$ 29-dominating sets can be enumerated with polynomial delay, and unit square graphs have locally bounded LMIM-width with

$\operatorname{imtw}(G)$ 30

which yields an incremental polynomial-time algorithm for enumerating minimal dominating sets (Høgemo et al., 2019, Golovach et al., 2015).

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

$\operatorname{imtw}(G)$ 31

satisfies

$\operatorname{imtw}(G)$ 32

for every cut. Chordal graphs and co-comparability graphs have sim-width at most $\operatorname{imtw}(G)$ 33, but their mim-width is unbounded in general; bounded mim-width reappears on $\operatorname{imtw}(G)$ 34-free chordal graphs and $\operatorname{imtw}(G)$ 35-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 $\operatorname{imtw}(G)$ 36, Independent Set and Feedback Vertex Set are solvable in time $\operatorname{imtw}(G)$ 37 (Kang et al., 2016, Bergougnoux et al., 2023).

6. Structural limits, classification results, and current directions

The decomposition-given nature of these parameters is a recurrent limitation. For induced matching treewidth, if $\operatorname{imtw}(G)$ 38, then one can compute in time $\operatorname{imtw}(G)$ 39 a tree decomposition $\operatorname{imtw}(G)$ 40 with

$\operatorname{imtw}(G)$ 41

but exact recognition is hard: for every fixed $\operatorname{imtw}(G)$ 42, deciding whether $\operatorname{imtw}(G)$ 43 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 $\operatorname{imtw}(G)$ 44, yet restricted classes such as $\operatorname{imtw}(G)$ 45-free graphs may still have unbounded induced matching treewidth (Lima et al., 2024).

On the mim-width side, hereditary classification has become a central topic. “On algorithmic applications of sim-width and mim-width of $\operatorname{imtw}(G)$ 46-free graphs” gives a complete dichotomy for the classes

$\operatorname{imtw}(G)$ 47

showing that bounded mim-width occurs exactly when $\operatorname{imtw}(G)$ 48 and one of $\operatorname{imtw}(G)$ 49 is at most $\operatorname{imtw}(G)$ 50, or $\operatorname{imtw}(G)$ 51 and one of $\operatorname{imtw}(G)$ 52 is at most $\operatorname{imtw}(G)$ 53. The same paper gives substantial partial classifications for

$\operatorname{imtw}(G)$ 54

graphs, and shows that bounded, quickly computable sim-width implies polynomial-time solvability of List $\operatorname{imtw}(G)$ 55-Colouring. It also establishes that if Independent Set is polynomial-time solvable on a class of bounded, quickly computable sim-width, then Independent $\operatorname{imtw}(G)$ 56-Packing is polynomial-time solvable there as well; this includes Induced Matching as the special case $\operatorname{imtw}(G)$ 57 (Munaro et al., 2022).

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 $\operatorname{imtw}(G)$ 58-vertex graph of sim-width $\operatorname{imtw}(G)$ 59 has neighbor-depth $\operatorname{imtw}(G)$ 60, implying a quasipolynomial-time algorithm for Independent Set on bounded sim-width, and therefore on bounded o-mim-width, even without a supplied decomposition (Bergougnoux et al., 2023).

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.