Split-by-edges trees

TL;DR

The paper introduces split-by-edges trees, a binary tree structure that encodes the independent sets of a graph, highlighting the relationship between tree leaves and maximum independent sets.

Contribution

It defines the split-by-edges tree structure and explores its properties relating to independent sets and maximum independent sets in graphs.

Findings

Every maximal independent set corresponds to a leaf in the tree.

Maximum independent sets are the closest leaves to the root.

The structure provides insights into the graph's independent sets.

Abstract

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} in N such that L = N - v, R = N - u, and uv is an edge in G. It follows from the definition that every maximal independent set in G is a leaf in T, and the maximum independent sets of G are the leaves closest to the root of T.