# On the semi-proper orientations of graphs

## TL;DR

This paper introduces the concept of semi-proper orientations in graphs, proves the existence of optimal orientations with weights of one or two, and establishes NP-hardness results for certain classes of graphs.

## Contribution

It defines the semi-proper orientation number, proves the existence of optimal orientations with limited weights, and shows NP-hardness for planar and bipartite graphs.

## Key findings

- Optimal semi-proper orientations exist with edge weights of one or two.
- Deciding if a planar graph with semi-proper orientation number 2 has an all-one edge orientation is NP-complete.
- Determining the semi-proper orientation number of planar bipartite graphs is NP-hard.

## 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_{v\in 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_{v\in 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.