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- W2944801434 abstract "A {it 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) neq S_{(D,w)}(u) $, where $S_{(D,w)}(v) $ is the sum of the weights of edges with head $v$ in $D$. The {it semi-proper orientation number} of a graph $G$, denoted by $overrightarrow{chi}_s (G)$, is $ min_{(D,w)in Gamma} max_{vin V(G)} S_{(D,w)}(v) $, where $Gamma$ is the set of all semi-proper orientations of $G$. The {it optimal semi-proper orientation} is a semi-proper orientation $(D,w)$ such that $ max_{vin V(G)} S_{(D,w)}(v)= overrightarrow{chi}_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 $overrightarrow{chi}_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." @default.
- W2944801434 created "2019-05-16" @default.
- W2944801434 creator A5064297618 @default.
- W2944801434 date "2019-05-08" @default.
- W2944801434 modified "2023-09-27" @default.
- W2944801434 title "On the semi-proper orientations of graphs" @default.
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