Depth-first search in split-by-edges trees

Abstract A split-by-edges tree of a graph G is a binary tree T where the root = V(G), every leaf is an independent set in G, and for every other node N in T with children L and R there is an edge uv in G such that u and v are in N, L = N – v and R = N – u. Every maximum independent set in G is in the same layer M, which is guarantied to be found in a layer by layer search of T. Every depth-first search in T will reach M, and may stop there, if a leaf was hit, or will pass through, to another, smaller, leaf below. For random graphs of a given order and increasing size the succes rates of depth-first searches in split-by-edges trees oscillate between local minima and maxima.