Fast algorithm of equitably partitioning degenerate graphs into graphs with lower degeneracy

Abstract

A q-degenerate k-partition of a graph G is a collection (V1,V2,…,Vk) of k pairwise disjoint subsets of V(G) such that V(G)=⋃i=1kVi and each Vi induces a q-degenerate subgraph. Such a partition is called equitable if ||Vi|−|Vj||≤1 for every 1≤i<j≤k. Equitable partition of graphs can model the problem of partitioning a large network into smaller sub-modules based on some cardinal principles, and has many other applications in network science and information science. In this work, we establish theoretical and algorithmic results on equitably partitioning degenerate graphs into graphs with lower degeneracy. Specifically, we show that every n-vertex d-degenerate graph with maximum degree at most n/β has a q-degenerate k-partition (V1,V2,…,Vk) for every k≥αd so that each Vi has size at most ⌈n/k⌉ whenever q,α, and β satisfy a well-defined inequality, and such a partition can be computed in cubic time.