Sandwich Monotonicity and the Recognition of Weighted Graph Classes

2025-08-08

AI Summary

This paper addresses the recognition problem for edge-weighted graphs whose *level graphs*—induced by all possible weight thresholds—belong to a fixed class of classical unweighted graphs, such as split graphs, threshold graphs, or chain graphs. We introduce the novel notion of *degree sandwich monotonicity*, which captures structural constraints across level graphs and enables a unified characterization framework. Methodologically, we integrate threshold-based edge elimination orderings, hierarchical level-graph structure analysis, and intrinsic degree-constraint monotonicity inherent to the target graph classes. This yields the first necessary and sufficient condition for linear-time recognizability of such weighted graph classes. Our algorithms achieve optimal *O(n + m)* time complexity for all three classes. Beyond recognition, our results provide new theoretical tools and algorithmic foundations for Robinson matrix recognition and similarity analysis, thereby extending both the structural theory and practical applicability of weighted graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Recognizing weighted graph classes with specific unweighted level graphs
• Introducing degree sandwich monotone graph classes for linear-time recognition
• Establishing conditions for efficient weighted graph class recognition algorithms

Innovation

• Linear-time recognition using elimination orderings
• Degree sandwich monotone graph classes introduced
• Necessary and sufficient conditions for algorithm