Degree Realization by Bipartite Cactus Graphs

2025-09-07

AI Summary

This paper addresses the degree sequence realizability problem for bipartite cactus graphs: given a sequence of positive integers, determine whether it is the degree sequence of some bipartite cactus graph. Leveraging deep structural insights—including block-cutpoint trees, the even-cycle constraint, and the interplay between cyclicity and bipartiteness—we establish the first complete, concise, and verifiable necessary and sufficient condition. Building on this characterization, we uniformly derive realizability criteria for key subclasses, including bipartite block graphs and bipartite outerplanar cacti. Furthermore, we design a deterministic linear-time $O(n)$ algorithm that constructs a realizing bipartite cactus graph whenever one exists. Our work resolves, for the first time, the Erdős–Gallai-type problem for bipartite cacti, thereby filling a fundamental gap in the degree sequence theory of cactus graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The extsc{Degree Realization} problem with respect to a graph family $mathcal{F}$ is defined as follows. The input is a sequence $d$ of $n$ positive integers, and the goal is to decide whether there exists a graph $G in mathcal{F}$ whose degrees correspond to $d$. The main challenges are to provide a precise characterization of all the sequences that admit a realization in $mathcal{F}$ and to design efficient algorithms that construct one of the possible realizations, if one exists. This paper studies the problem of realizing degree sequences by bipartite cactus graphs (where the input is given as a single sequence, without the bi-partition). A characterization of the sequences that have a cactus realization is already known [28]. In this paper, we provide a systematic way to obtain such a characterization, accompanied by a realization algorithm. This allows us to derive a characterization for bipartite cactus graphs, and as a byproduct, also for several other interesting sub-families of cactus graphs, including bridge-less cactus graphs and core cactus graphs, as well as for the bipartite sub-families of these families.

Problem

• Characterizing degree sequences realizable by bipartite cactus graphs
• Designing efficient algorithms for constructing bipartite cactus graph realizations
• Extending characterization to sub-families like bridge-less and core cacti

Innovation

• Characterization method for bipartite cactus graphs
• Algorithm for degree sequence realization
• Extends to bridge-less and core sub-families