The induced matching treewidth of a graph 
is the minimum, over all tree
decompositions, of the largest size of an induced
matching that has at least one endpoint of every edge
in a single bag. The induced matching is taken
in 
,
so its other endpoints need not lie in the bag (Alon et al. 2026).
Writing this parameter as 
gives
 |
where the right side is the tree-independence number. The inequality follows by choosing one endpoint in the bag from each edge of the induced matching.
An empty graph has value 0. Every complete bipartite graph with both parts nonempty has value 1, whereas the tree-independence
number of 
is unbounded. For graphs excluding a fixed induced complete
bipartite graph, Alon et al. (2026) prove a polynomial
upper bound in the reverse direction.
