This short paper treats some decision problems for Petri Nets and shows that the problems of the safeness, the reachability and the proper termination are NP-complete for Petri Nets in which each transition fires at most once.

Let G = (A, B, E) be a bipartite graph with bipartition (A:B) of vertex set and edge set E. For each vertex v, Γ(v) denotes the set of adjacent vertices to v. G is said to be t-convex on the vertex set A if there is a tree and a one-to-one correspondence between vertices in A and edges of the tree such that for each vertex b B the edges of the tree corresponding to vertices in Γ(b) form a path on the tree. G is doubly t-convex if it is convex both on vertex set A and on B. In this paper, we show that, the class of doubly t-convex graphs is exactly the class of bipartite circle graphs.