Tree Covers of Graphs and their Linear Preservers

Let G be an undirected graph. A \textit{vertex tree cover} of G is a collection of trees such that every vertex of G is incident with at least one tree. Similarly, an edge tree cover is a collection of trees such that every edge of G is contained in at least one tree. The tree cover number is defined as the minimum number of trees required in such a cover. In this paper, we demonstrate that when considering specific types of tree covers, only vertex permutations act as linear operators that preserve the tree cover number of G .