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Reports Year : 2022

## Semi-proper orientations of dense graphs

Julio Araujo
• Function : Author
Frédéric Havet
Nicolas Nisse

#### Abstract

An {\it 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 {\it $k$-orientation} if the in-degree of each vertex in $D$ is at most $k$. An orientation $D$ of $G$ is {\it proper} if any two adjacent vertices have different in-degrees in $D$. The {\it proper orientation number} of a graph $G$, denoted by $\po(G)$, is the minimum $k$ such that $G$ has a proper $k$-orientation. A {\it weighted orientation} of a graph $G$ is a pair $(D,w)$, where $D$ is an orientation of $G$ and $w$ is an arc-weighting $A(D) \ra \mathbb{N}\setminus\{0\}$. A {\it 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 $S_{(D,w)}(v) \neq S_{(D,w)}(u)$, where $S_{(D,w)}(v)$ is the sum of the weights of the arcs in $(D,w)$ with head $v$. For a positive integer $k$, a {\it semi-proper $k$-orientation} $(D,w)$ of a graph $G$ is a semi-proper orientation of $G$ such that $\max_{v\in V(G)} S_{(D,w)}(v) \leq k$. The {\it semi-proper orientation} number of a graph $G$, denoted by $\spo(G)$, is the least $k$ such that $G$ has a semi-proper $k$-orientation. In this work, we first prove that $\spo(G) \in \{\omega(G)-1,\omega(G)\}$ for every split graph $G$, and that, given a split graph $G$, deciding whether $\spo(G) = \omega(G)-1$ is an $\NP$-complete problem. We also show that, for every $k$, there exists a (chordal) graph $G$ and a split subgraph $H$ of $G$ such that $\po(G) \leq k$ and $\po(H) = 2k-2$. In the sequel, we show that, for every $n\geq p(p+1)$, $\spo(P^{p}_n) = \left\lceil \frac{3}{2}p \right\rceil$, where $P^{p}_n$ is the $p^{th}$ power of the path on $n$ vertices. We investigate further unit interval graphs with no big clique: we show that $\po(G) \leq 3$ for any unit interval graph $G$ with $\omega(G)=3$, and present a complete characterization of unit interval graphs with $\po(G)=\omega(G)=3$. Then, we show that deciding whether $\spo(G)=\omega(G)$ can be solved in polynomial time in the class of co-bipartite graphs. Finally, we prove that computing $\spo(G)$ is $\FPT$ 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 $\spo(G)$, but also $\po(G)$, admits a polynomial kernel when parameterized by the neighbourhood diversity plus the value of the solution. These results imply kernels of size $4^{{\cal O}(k^2)}$ and ${\cal O}(2^kk^2)$, in chordal graphs and split graphs, respectively, for the problem of deciding whether $\spo(G)\leq k$ parameterized by $k$. We also present exponential kernels for computing both $\po(G)$ and $\spo(G)$ parameterized by the value of the solution when $G$ is a cograph. On the other hand, we show that computing $\spo(G)$ does not admit a polynomial kernel parameterized by the value of the solution when $G$ is a chordal graph, unless $\NP \subseteq \coNP/\poly$.

### Dates and versions

hal-03907202 , version 1 (19-12-2022)
hal-03907202 , version 2 (13-01-2023)

### Identifiers

• HAL Id : hal-03907202 , version 2

### Cite

Julio Araujo, Frédéric Havet, Claudia Linhares Sales, Nicolas Nisse, Karol Suchan. Semi-proper orientations of dense graphs. Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France. 2022. ⟨hal-03907202v2⟩

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