Sandwich Monotonicity and the Recognition of Weighted Graph Classes

TL;DR

This paper introduces the concept of degree sandwich monotone graph classes and demonstrates linear-time recognition algorithms for weighted graphs whose level graphs belong to such classes, expanding understanding of graph recognition.

Contribution

It defines degree sandwich monotone classes and provides necessary and sufficient conditions for linear-time recognition of weighted graphs within these classes.

Findings

Recognition algorithms for split, threshold, and chain graphs in linear time.

Introduction of degree sandwich monotone graph classes.

Characterization of conditions for linear-time recognition.

Abstract

Edge-weighted graphs play an important role in the theory of Robinsonian matrices and similarity theory, particularly via the concept of level graphs, that is, graphs obtained from an edge-weighted graph by removing all sufficiently light edges. This suggest a natural way of associating to any class $\mathcal{G}$ of unweighted graphs a corresponding class of edge-weighted graphs, namely by requiring that all level graphs belong to $\mathcal{G}$. We show that weighted graphs for which all level graphs are split, threshold, or chain graphs can be recognized in linear time using special edge elimination orderings. We obtain these results by introducing the notion of degree sandwich monotone graph classes. A graph class $\mathcal{G}$ is sandwich monotone if every edge set which may be removed from a graph in $\mathcal{G}$ without leaving the class also contains a single edge that can be safely removed. Furthermore, if we require the safe edge to fulfill a certain degree property, then $\mathcal{G}$ is called degree sandwich monotone. We present necessary and sufficient conditions for the existence of a linear-time recognition algorithm for any weighted graph class whose corresponding unweighted class is degree sandwich monotone and contains all edgeless graphs.