On upper domatic number of graphs

Let $G=(V,E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ $\mathit{\text{dominates}}$ $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $\pi =\{V_1,V_2,\dots ,V_k\}$ of $G$ is called an \emph{upper domatic partition} of size $k$ if either $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for all $i,j$, where $1\le i<j\le k$. The maximum integer $k$ for which the above partition exists is called the \emph{upper domatic number} of $G$, and it is denoted by $D(G)$. The $\text{textsc}MaximumupperdomaticnumberProblem$ involves finding an upper domatic partition of a given graph with the maximum number of parts. It was known that the maximum upper domatic problem can be solved in linear time for trees. In this paper, we prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. Moreover, we show that this problem can be solved in polynomial time for unicyclic graphs. Finally, we partially solve a conjecture regarding the sink set posed by Haynes et al. [The upper domatic number of a graph, \emph{AKCE Int. J. Graphs Comb.}, 17, 2020].