On the semi-proper orientations of graphs
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A semi-proper orientation of a given graph G is a function (D,w) that assigns an orientation D(e) and a positive integer weight w(e) to each edge e such that for every two adjacent vertices v and u, S_(D,w)(v) ≠ S_(D,w)(u) , where S_(D,w)(v) is the sum of the weights of edges with head v in D. The semi-proper orientation number of a graph G, denoted by χ_s (G), is _(D,w)∈Γ_v∈ V(G) S_(D,w)(v) , where Γ is the set of all semi-proper orientations of G. The optimal semi-proper orientation is a semi-proper orientation (D,w) such that _v∈ V(G) S_(D,w)(v)= χ_s (G) . In this work, we show that every graph G has an optimal semi-proper orientation (D,w) such that the weight of each edge is one or two. Next, we show that determining whether a given planar graph G with χ_s (G)=2 has an optimal semi-proper orientation (D,w) such that the weight of each edge is one is NP-complete. Finally, we prove that the problem of determining the semi-proper orientation number of planar bipartite graphs is NP-hard.
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