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    Topological Additive Numbering of Directed Acyclic Graphs

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    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 DD be a digraph and ff a labeling of its vertices with positive integers; denote by S(v)S(v) the sum of labels over all neighbors of each vertex vv. The labeling ff is called \emph{topological additive numbering} if S(u)<S(v)S(u) < S(v) for each arc (u,v)(u,v) of the digraph. The problem asks to find the minimum number kk for which DD has a topological additive numbering with labels belonging to {1,…,k}\{ 1, \ldots, k \}, denoted by ηt(D)\eta_t(D). We characterize when a digraph has topological additive numberings, give a lower bound for ηt(D)\eta_t(D), and provide an integer programming formulation for our problem, characterizing when its coefficient matrix is totally unimodular. We also present some families for which ηt(D)\eta_t(D) 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
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