The POC Orientation Number on Vertex Weighted Spiders

Let (G, w) be a vertex weighted graph such that w:V(G) → {1,2,...}. A vertex coloring c:V(G) → {0,1,...} is a properly ordered coloring (POC) if for any edge e=uv in G, c(u) > c(v) whenever w(u) > w(v) and c(u) ≠ c(v) whenever w(u)=w(v). An orientation D on (G, w) such that the indegrees of D forms a POC on (G, w) is called a POC orientation. In this paper, we give a characterization for vertex weighted trees (T, w) with a specific restriction and w:V(T) → {1, 2} such that there exists a POC orientation on (T, w).