Threshold graph

2020 Mathematics Subject Classification: Primary: 05C [MSN][ZBL]

A graph with Dilworth number $1$: for any two vertices $x,y$, one of the neighbourhoods $N(x)$, $N(y)$ contains the other. Such graphs are characterised by having no induced subgraph of the form $K_{2,2}$ (complete bipartite on $2+2$ points) , $C_4$ (cycle of length $4$) or $P_4$ (path of length $4$). They are the comparability graphs of threshold orders.

There is a polynomial time algorithm for computing the Dilworth number of a finite graph.

References

• Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications 3. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 Zbl 0919.05001