On the threshold-width of graphs

TL;DR

This paper introduces the concept of threshold-width for graphs, proves its NP-completeness, and provides fixed-parameter algorithms along with characterizations via forbidden subgraphs.

Contribution

It defines threshold-width, establishes its computational complexity, and offers algorithms and structural characterizations for graphs with bounded threshold-width.

Findings

TH-width decision problem is NP-complete.

Fixed-parameter algorithms are developed for TH-width.

Graphs with bounded TH-width are characterized by finite forbidden subgraphs.

Abstract

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs.