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Topological Additive Numbering of Directed Acyclic Graphs
We propose to study a problem that arises naturally from both Topological
Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as
Lucky Labeling). Let be a digraph and a labeling of its vertices with
positive integers; denote by the sum of labels over all neighbors of
each vertex . The labeling is called \emph{topological additive
numbering} if for each arc of the digraph. The problem
asks to find the minimum number for which has a topological additive
numbering with labels belonging to , denoted by
.
We characterize when a digraph has topological additive numberings, give a
lower bound for , and provide an integer programming formulation for
our problem, characterizing when its coefficient matrix is totally unimodular.
We also present some families for which can be computed in
polynomial time. Finally, we prove that this problem is \np-Hard even when its
input is restricted to planar bipartite digraphs