I am trying to draw a reachability graph for this Petri net. Could you please help me to understand how this works?
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The nodes of the reachability graph of a Petri net consists of markings of the net, and one connects them with edges that represent the firing of a transition.
In your example start with a node representing the initial marking $(5,0,3,2,0)$. In this marking transition $T0$ taking $1$ from places $P0$ and $P2$ each, while adding $1$ to place $P1$. The resulting marking is $(4,1,2,2,0)$. Add an edge labelled by $T0$ from the first to second marking.
In the same initial marking also $T2$ is enabled, add an edge to the marking that results after firing. Continue this way, visiting the newly generated markings, until all possibilities are taken care of.
If the net is bounded, the number of reachable markings is finite, and your construction finishes in a finite number of steps.