  Petri nets proved useful to describe various real-world systems, but many of
their properties are very hard to check. To alleviate this difficulty,
subclasses are often considered. The class of weighted marked graphs with
relaxed place constraint (WMG=< for short), in which each place has at most one
input and one output, and the larger class of choice-free (CF) nets, in which
each place has at most one output, have been extensively studied to this end,
with various applications.
  In this work, we develop new properties related to the fundamental and
intractable problems of reachability, liveness and reversibility in weighted
Petri nets. We focus mainly on the homogeneous Petri nets with a single shared
place (H1S nets for short), which extend the expressiveness of CF nets by
allowing one shared place (i.e. a place with at least two outputs and possibly
several inputs) under the homogeneity constraint (i.e. all the output weights
of the shared place are equal). Indeed, this simple generalization already
yields new challenging problems and is expressive enough for modeling existing
use-cases, justifying a dedicated study.
  One of our central results is the first characterization of liveness in a
subclass of H1S nets more expressive than WMG=< that is expressed by the
infeasibility of an integer linear program (ILP) of polynomial size. This trims
down the complexity to co-NP, contrasting with the known EXPSPACE-hardness of
liveness in the more general case of weighted Petri nets. In the same subclass,
we obtain a new reachability property related to the live markings, which is a
variant of the well-known Keller's theorem. Another central result is a new
reversibility characterization for the live H1S class, simplifying its
checking. Finally, we apply our results to use-cases, highlight their
scalability and discuss their extensibility to more expressive classes.
