Split-by-edges trees

A split-by-edges tree of a graph G on n vertices 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 a pair of vertices {u, v}  N such that L = N \ {v}, R = N \ {u}, and uv is an edge in G. The distance from the root to an independent set I is n – |I| and the maximum independent sets of G are the ones closest to the root. For every independent set X of G there is a leaf Y in T such that X  Y, thus every maximal independent set in G is a leaf in T. In a uniquified split-byedges tree a maximum independent set is found in a layer-by-layer search in at the most 2 (n) time, in terms of number of split operations, whith (n) = O(0.369425n) for random graphs. The SBE-tree An independent set I in a graph G = (V, E) is a set of vertices no two of which are adjacent, and if G has no larger independent set then I is a maximum independent set of G. Tarjan and Trojanowski point out that, given a vertex v  V, any maximum independent set of G must be a subset of either V – N(v) or V – v, using that as the starting point for an algorithm that finds a maximum independent set in less than 2 time [1]. Relatedly, given an edge uv  E and a maximum independent set M  V, either M  V \ {u} or M  V \ {v}. This gives rise to the following definition. Definition 1: Let G be graph and let T be a binary tree of subsets of V(G). Then T is a split-by-edges tree, or SBE-tree, of G if and only if the root of T = V(G), every leaf in T is an independent set of G, and for every other node N in T with children L and R there is a pair of vertices {u, v}  N such that L = N \ {u}, R = N \ {v}, and u and v are adjacent in G. Figure 1. An SBE-tree of the graph at the upper left. The leaves have bold blue frames. The gray nodes are duplicates of others. The branching labels, which do not belong to the tree, show the splitting edges. Theorem 1. Given a graph G and a split-by-edges tree T of G, for every independent set X of G there is a leaf Y in T such that X  Y. Proof: Given an independent set I of G, for every node N in T, if I  N then I  L(N) or I  L(N).  Corollary 1.1. Every maximal independent set of G is a leaf in T. 