MAX for $k$-independence in multigraphs

TL;DR

This paper analyzes the worst-case size of $k$-independent sets produced by the MAX algorithm in multigraphs, providing an efficient method to determine the minimal possible size based on degree sequences.

Contribution

It introduces a procedure to compute the smallest $k$-independent set size achievable by MAX for any degree sequence, establishing sharp bounds for multigraphs.

Findings

The procedure accurately determines the minimal $k$-independent set size for given degree sequences.

The analysis is sharp, with examples showing the bounds are tight.

Provides insights into the behavior of MAX in multigraphs with specified degree sequences.

Abstract

For a fixed positive integer $k$, a set $S$ of vertices of a graph or multigraph is called a $k$-independent set if the subgraph induced by $S$ has maximum degree less than $k$. The well-known algorithm MAX finds a maximal $k$-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than $k$. We give an efficient procedure that determines, for a given degree sequence $D$, the smallest cardinality $b(D)$ of a $k$-independent set that can result from any application of MAX to any loopless multigraph with degree sequence $D$. This analysis of the worst case is sharp for each degree sequence $D$ in that there exists a multigraph $G$ with degree sequence $D$ such that some application of MAX to $G$ will result in a $k$-independent set of cardinality exactly $b(D)$.