On the proper orientation number of chordal graphs

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is proper if $d^-_D(u)\neq d^-_D(v)$, for all $uv\in E(G)$. An orientation with maximum indegree at most $k$ is called a $k$-orientation. The proper orientation number of $G$, denoted by $\overrightarrow{\chi}(G)$, is the minimum integer $k$ such that $G$ admits a proper $k$-orientation. We prove that determining whether $\overrightarrow{\chi}(G) \leq k$ is NP-complete for chordal graphs of bounded diameter, but can be solved in linear-time in the subclass of quasi-threshold graphs. When parameterizing by $k$, we argue that this problem is FPT for chordal graphs and argue that no polynomial kernel exists, unless $NP\subseteq coNP/\ poly$. We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs. Concerning bounds, we prove tight upper bounds for subclasses of block graphs. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph $G$ having no adjacent vertices of degree at least $c+1$ have proper orientation number at most $c$. This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs $G$ whose weak-dual is a path satisfy $\overrightarrow{\chi}(G)\leq 13$. Finally, we present simple bounds to the classes of chordal claw-free graphs and cographs.