  We address the following general question: given a graph class C on which we
can solve Maximum Matching in (quasi) linear time, does the same hold true for
the class of graphs that can be modularly decomposed into C ? A major
difficulty in this task is that the Maximum Matching problem is not preserved
by quotient, thereby making difficult to exploit the structural properties of
the quotient subgraphs of the modular decomposition. So far, we are only aware
of a recent framework in [Coudert et al., SODA'18] that only applies when the
quotient subgraphs have bounded order and/or under additional assumptions on
the nontriv-ial modules in the graph. As a first attempt toward improving this
framework we study the combined effect of modular decomposition with a pruning
process over the quotient subgraphs. More precisely, we remove sequentially
from all such subgraphs their so-called one-vertex extensions (i.e., pendant,
anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a
"pruned modular decomposition", that can be computed in O(m log n)-time. Our
main result is that if all the pruned quotient subgraphs have bounded order
then a maximum matching can be computed in linear time. This result is mostly
based on two pruning rules on pendant and anti-pendant modules -- that are
adjacent, respectively, to one or all but one other modules in the graph.
Furthermore, these two latter rules are surprisingly intricate and we consider
them as our main technical contribution in the paper. We stress that the class
of graphs that can be totally decomposed by the pruned modular decomposition
contains all the distance-hereditary graphs, and so, it is larger than
cographs. In particular, as a byproduct of our approach we also obtain the
first known linear-time algorithms for Maximum Matching on distance-hereditary
graphs and graphs with modular-treewidth at most one. Finally, we can use an
extended version of our framework in order to compute a maximum matching, in
linear-time, for all graph classes that can be modularly decomposed into
cycles. Our work is the first to explain why the existence of some nice
ordering over the modules of a graph, instead of just over its vertices, can
help to speed up the computation of maximum matchings on some graph classes.
