Partition of graphs with maximum degree ratio

TL;DR

This paper introduces a new measure called the degree ratio for graph partitions, analyzes its bounds and exact values for certain graph classes, and explores related computational complexity issues.

Contribution

It defines the degree ratio of a graph, provides bounds and exact values for specific classes, and investigates the complexity of related problems.

Findings

Bounds and exact values of q(G) for some graph classes

Complexity results for optimization and decision problems related to degree ratio

Introduction of a new graph partition quality measure

Abstract

Given a graph $G$ and a non trivial partition $(V_1,V_2)$ of its vertex-set, the satisfaction of a vertex $v\in V_i$ is the ratio between the size of it's closed neighborhood in $V_i$ and the size of its closed neighborhood in $G$. The worst ratio over all the vertices defines the quality of the partition. We define $q(G)$ the degree ratio of a graph as the maximum of the worst ratio over all the non trivial partitions. We give bounds and exact values of $q(G)$ for some classes of graphs. We also show some complexity results for the associated optimization or decision problems.