Complexity of List Homomorphisms to Separable Signed Graphs

Core Concepts

The complexity of the list homomorphism problem for separable signed graphs is classified, showing that it is polynomial-time solvable for segmented signed graphs and NP-complete otherwise.

Abstract

The paper investigates the complexity of list homomorphism problems for signed graphs, focusing on a special class called separable signed graphs. Separable signed graphs are irreflexive signed graphs in which the unicoloured edges form a spanning path or cycle. The key findings are: For path-separable signed graphs: If the signed graph is segmented (either right-segmented, left-segmented, or left-right-segmented), then the list homomorphism problem is polynomial-time solvable. Otherwise, the list homomorphism problem is NP-complete. For cycle-separable signed graphs: If the signed graph is semi-balanced, then the list homomorphism problem is polynomial-time solvable if there is a special min ordering, and NP-complete otherwise. If the signed graph is not semi-balanced, then the list homomorphism problem is NP-complete. The authors believe that the case of separable signed graphs, together with the case of irreflexive signed trees, will play an important role in the general classification of complexity for irreflexive signed graphs.

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List homomorphisms to separable signed graphs

by Jan ... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2306.06449.pdf

Deeper Inquiries

How can the insights from the analysis of separable signed graphs be extended to a more general classification of complexity for irreflexive signed graphs

The insights gained from the analysis of separable signed graphs can be extended to provide a more general classification of complexity for irreflexive signed graphs. By understanding the structural properties and characteristics of separable signed graphs, we can identify key patterns and relationships that may apply to a broader range of irreflexive signed graphs. For example, the identification of special cases where list homomorphisms are polynomial-time solvable in separable signed graphs can serve as a foundation for developing similar classifications for other types of irreflexive signed graphs. By examining the complexities of different subclasses of signed graphs, we can potentially establish a more comprehensive framework for understanding the computational challenges and solutions in this area.

What are the potential applications of the polynomial-time solvable cases of list homomorphisms to separable signed graphs

The polynomial-time solvable cases of list homomorphisms to separable signed graphs have various potential applications in graph theory and computational algorithms. Graph Coloring: The results obtained from studying list homomorphisms to separable signed graphs can contribute to the development of efficient graph coloring algorithms. Understanding the polynomial cases can lead to improved strategies for assigning colors to vertices in a way that satisfies certain constraints, which is crucial in various graph optimization problems. Network Design: In network design and optimization, the ability to efficiently solve list homomorphism problems for separable signed graphs can aid in designing robust and efficient network structures. By leveraging the polynomial-time solvability of certain cases, network engineers can make informed decisions about connectivity and resource allocation in complex networks. Constraint Satisfaction: The insights from polynomial cases of list homomorphisms can be applied to constraint satisfaction problems in various domains. By understanding the tractable instances of list homomorphisms in separable signed graphs, researchers and practitioners can develop better approaches for solving constraint satisfaction problems with specific structural constraints.

Are there any connections between the complexity of list homomorphisms to separable signed graphs and other graph-theoretic or computational problems

There are several connections between the complexity of list homomorphisms to separable signed graphs and other graph-theoretic or computational problems: Graph Isomorphism: The complexity of list homomorphisms to separable signed graphs can be linked to the graph isomorphism problem. Understanding the polynomial-time solvable cases in separable signed graphs may provide insights into the structural similarities and differences between isomorphic graphs, contributing to the development of efficient graph isomorphism algorithms. Constraint Satisfaction Problems: The study of list homomorphisms in separable signed graphs is closely related to constraint satisfaction problems. The polynomial-time solvable instances can offer valuable information on how constraints can be satisfied in specific graph structures, which can be applied to a wide range of constraint satisfaction problems in computer science and mathematics. Computational Complexity Theory: The analysis of list homomorphisms to separable signed graphs contributes to the broader field of computational complexity theory. By investigating the complexities of different graph problems, researchers can gain insights into the inherent computational challenges and possibilities within graph theory and related computational domains.