On a simple characterisation of threshold graphs

Abstract

In their very interesting paper “Set-packing problems and threshold graphs” [1] V. Chvatal and P. L. Hammer have shown that the constraints

$$\begin{gathered} \sum\limits_{j = 1}^n {a_{ij} x_j \leqslant 1(i = 1,2, . . . , m)} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$

are equivalent to the only inequality

$$\begin{gathered} \sum\limits_{j - 1}^n {c_j x_j \leqslant d} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$

if and only if the intersection graph associated with the matrix (a _ij ) — see § 1 — is a threshold graph i.e. a graph none of whose induced subgraphs are isomorphic to 2K ₂,P ₄,C ₄:

As Chvatal and Hammer have shown [1], threshold graphs can be characterised in many different ways; the main result of this paper is to give a new, very simple characterisation which will enable us to test whether a graph is a threshold by a simple inspection of its incidence matrix.