tree domatic number in graphs

A 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 \(3\).