Title

Weakly threshold graphs

Document Type

Article

Date of Original Version

1-1-2018

Abstract

We define a weakly threshold sequence to be a degree sequence d = (d1, ⋯, dn) of a graph having the property that Σi≤kdi ≥ k(k - 1) + Σi>k min {k, di} - 1 for all positive k ≤ max {i : di ≥ i - 1}. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs.

Publication Title

Discrete Mathematics and Theoretical Computer Science

Volume

20

Issue

1

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