Graphs with the Strong Havel–Hakimi Property

The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, 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 pro...

Full description

Saved in:
Bibliographic Details
Published inGraphs and combinatorics Vol. 32; no. 5; pp. 1689 - 1697
Main Authors Barrus, Michael D., Molnar, Grant
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.09.2016
Subjects
Combinatorics
Engineering Design
Mathematics
Mathematics and Statistics
Original Paper
05C07
05C69
Havel–Hakimi algorithm
Residue
Independence number
Online AccessGet full text
ISSN0911-0119
1435-5914
DOI10.1007/s00373-015-1674-7

Cover

More Information
Summary:The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, 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.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-015-1674-7