Tree domatic number in graphs

Abstract

Dominating set $S$ in a graph $G$ is a tree dominating set of $G$ if the subgraph induced by $S$ is a tree. The tree domatic number of $G$ is the maximum number of pairwise disjoint tree dominating sets in $V(G)$. First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most $4$ and give a characterization of planar graphs with the tree domatic number $3$.