The Critical Independence Number and an Independence Decomposition

TL;DR

This paper introduces the critical independence number, a new graph invariant that provides a polynomial-time computable lower bound for the independence number and enables a graph decomposition related to K"{o}nig-Egervary graphs.

Contribution

It defines the critical independence number, proves a decomposition theorem for graphs based on this number, and characterizes K"{o}nig-Egervary graphs through this concept.

Findings

Critical independence number is a polynomial-time computable lower bound.

Graphs can be decomposed into subgraphs with specific independence properties.

K"{o}nig-Egervary graphs are characterized by equality of independence and critical independence numbers.

Abstract

An independent set $I_c$ is a \textit{critical independent set} if $|I_c| - |N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical independence number} of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the subgraphs is the independence number of the graph. A proof of a conjecture of Graffiti.pc yields a new characterization of K\"{o}nig-Egervary graphs: these are exactly the graphs whose independence and critical independence numbers are equal.