Semi-proper orientations of dense graphs

An orientation $D$ of a graph $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation $D$ of a graph $G$ is a $k$-orientation if the in-degree of each vertex in $D$ is at most $k$. An orientation $D$ of $G$ is proper if any two adjacent vertices have different in-degrees in $D$. The proper orientation number of a graph $G$, denoted by $\vec{\chi }\left(G\right)$, is the minimum $k$ such that $G$ has a proper $k$-orientation.

A weighted orientation of a graph $G$ is a pair $(D,w)$, where $D$ is an orientation of $G$ and $w$ is an arc-weighting $A\left(D\right)\to N\setminus \left\{0\right\}$. A semi-proper orientation of $G$ is a weighted orientation $(D,w)$ of $G$ such that for every two adjacent vertices $u$ and $v$ in $G$, we have that $S_{(D,w)}\left(v\right)\ne S_{(D,w)}\left(u\right)$, where $S_{(D,w)}\left(v\right)$ is the sum of the weights of the arcs in $(D,w)$ with head $v$. For a positive integer $k$, a semi-proper $k$-orientation $(D,w)$ of a graph $G$ is a semi-proper orientation of $G$ such that ${max}_{v\in V\left(G\right)}{S}_{(D,w)}\left(v\right)\le k$. The semi-proper orientation number of a graph $G$, denoted by $\overrightarrow{{\chi }_s}\left(G\right)$, is the least $k$ such that $G$ has a semi-proper $k$-orientation.

In this work, we first prove that $\overrightarrow{{\chi }_s}\left(G\right)\in \{\omega \left(G\right)-1,\omega \left(G\right)\}$ for every split graph $G$, and that, given a split graph $G$, deciding whether $\overrightarrow{{\chi }_s}\left(G\right)=\omega \left(G\right)-1$ is an $\mathrm{NP}$-complete problem. We also show that, for every $k$, there exist a (chordal) graph $G$ and a split subgraph $H$ of $G$ such that $\vec{\chi }\left(G\right)\le k$ and $\vec{\chi }\left(H\right)=2k-2$. In the sequel, we show that, for every $n\ge p(p+1)$, $\overrightarrow{{\chi }_s}\left(P_n^p\right)=\left(\frac{3}{2}p\right)$, where $P_n^p$ is the $p$th power of the path on $n$ vertices. We investigate further unit interval graphs with no big clique: we show that $\vec{\chi }\left(G\right)\le 3$ for any unit interval graph $G$ with $\omega \left(G\right)=3$, and present a complete characterization of unit interval graphs with $\vec{\chi }\left(G\right)=\omega \left(G\right)=3$. Then, we show that deciding whether $\overrightarrow{{\chi }_s}\left(G\right)=\omega \left(G\right)-1$ can be solved in polynomial time in the class of cobipartite graphs. Finally, we prove that computing $\overrightarrow{{\chi }_s}\left(G\right)$ is $\mathrm{FPT}$ when parameterized by the minimum size of a vertex cover in $G$ or by the treewidth of $G$. We also prove that not only computing $\overrightarrow{{\chi }_s}\left(G\right)$, but also $\vec{\chi }\left(G\right)$, admits a polynomial kernel when parameterized by the neighborhood diversity plus the value of the solution. These results imply kernels of size $4^{O\left(k^2\right)}$ and $O\left(2^k{k}^2\right)$, in chordal graphs and split graphs, respectively, for the problem of deciding whether $\overrightarrow{{\chi }_s}\left(G\right)\le k$ parameterized by $k$. We also present exponential kernels for computing both $\vec{\chi }\left(G\right)$ and $\overrightarrow{{\chi }_s}\left(G\right)$ parameterized by the value of the solution when $G$ is a cograph. On the other hand, we show that computing $\overrightarrow{{\chi }_s}\left(G\right)$ does not admit a polynomial kernel parameterized by the value of the solution when $G$ is a chordal graph, unless $\mathrm{NP}\subseteq \mathrm{coNP}/\mathrm{poly}$.