Splitting trees

Splitting a tree is defined as removing all edges of a chain and disconnecting one from the other edges incident with that chain. Splitting a forest is simultaneously splitting each of its non-trivial trees. The splitting number a(T) of a tree T is the minimum number of successive forest splittings which lead to deletion of all of T’s edges. An O(N) algorithm is proposed to get an upper bound c’(t), the connected splitting number, on the splitting number o(t) of a tree T and an O(N log N) algorithm to compute this last number, where N is the number of vertices of the tree. Subject to a mild condition, these numbers lead to find a ‘black-and-white coloring’ of a tree T. In such a coloring a large part of T’s vertices are colored in black or white and no two adjacent vertices receive a different color.