Graphs with the strong Havel-Hakimi property

TL;DR

This paper characterizes graphs with the strong Havel-Hakimi property, showing they include threshold and matrogenic graphs, and demonstrates that for these graphs, the residue equals the independence number, with a greedy algorithm finding maximum independent sets.

Contribution

It provides a characterization of graphs with the strong Havel-Hakimi property via minimal forbidden induced subgraphs and links the residue to the independence number.

Findings

Graphs with the strong Havel-Hakimi property include threshold and matrogenic graphs.

For these graphs, the residue equals the independence number.

A natural greedy algorithm always finds a maximum independent set.

Abstract

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.