Two faces of greedy leaf removal procedure on graphs

TL;DR

This paper provides an analytical study of the roots in the greedy leaf removal procedure on random graphs, offering a unified mean-field theory that explains the sizes of minimal vertex covers and maximum matchings.

Contribution

It introduces a simple geometrical interpretation and a concise mean-field theory for roots in the GLR procedure, extending understanding beyond Erdős-Rényi graphs.

Findings

Analytical results for roots in the GLR procedure on random graphs.

Reproduction of zero-temperature replica symmetric estimates for vertex covers and matchings.

Unified theory applicable to graphs with or without cores.

Abstract

The greedy leaf removal (GLR) procedure on a graph is an iterative removal of any vertex with degree one (leaf) along with its nearest neighbor (root). Its result has two faces: a residual subgraph as a core, and a set of removed roots. While the emergence of cores on uncorrelated random graphs was solved analytically, a theory for roots is ignored except in the case of Erd\"{o}s-R\'{e}nyi random graphs. Here we analytically study roots on random graphs. We further show that, with a simple geometrical interpretation and a concise mean-field theory of the GLR procedure, we reproduce the zero-temperature replica symmetric estimation of relative sizes of both minimal vertex covers and maximum matchings on random graphs with or without cores.