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Extremal Minimal Bipartite Matching Covered Graphs: A Complete Characterization


Core Concepts
Every extremal minimal bipartite matching covered graph is obtained from two copies of a halin tree by adding a matching between the corresponding leaves.
Abstract
The paper provides a complete characterization of the class of extremal minimal bipartite matching covered graphs. Key highlights: Bipartite graphs are matching covered if and only if they admit an ear decomposition (Hetyei's theorem). Lov??sz and Plummer proved that minimal bipartite matching covered graphs have at least 2(m-n+2) vertices of degree two. The authors define five notions of extremality based on different lower bounds on the number of 2-edges and 2-vertices. The main result is a characterization of 2-vertex extremal minimal bipartite matching covered graphs (H2): they are obtained from two copies of a halin tree by adding a matching between the corresponding leaves. The authors also characterize the other extremal classes (H3, H4, H0, H1) using similar techniques, relating them to the main characterization. The authors establish relationships between the different extremal classes and provide a containment poset.
Stats
Every extremal minimal bipartite matching covered graph, distinct from C4, has: |V3| = 3n - 2m - 4 |E3,2| = 2m - 2n + 4 |E3| = 3n - 2m - 6 = |V3| - 2
Quotes
"A graph is a matching covered if every edge is present in some perfect matching." "Every extremal minimal bipartite matching covered graph, distinct from C4, satisfies the balanced 2-cut property."

Key Insights Distilled From

by Amit Kumar M... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06445.pdf
Extremal minimal bipartite matching covered graphs

Deeper Inquiries

How can the characterizations of the extremal classes be extended to the non-extremal minimal bipartite matching covered graphs?

The characterizations of the extremal classes can be extended to non-extremal minimal bipartite matching covered graphs by considering the properties and structural characteristics identified in the extremal classes. By understanding how extremal graphs differ from non-extremal ones, we can infer certain patterns or features that may still hold true in the non-extremal cases. For example, the lower bounds on the number of 2-edges or the presence of balanced 2-cuts in extremal graphs may provide insights into the structure of non-extremal graphs as well. By analyzing the relationships between extremal and non-extremal graphs, we can potentially identify common traits or properties that are shared across both categories.

What are the implications of the balanced 2-cut property beyond the extremal classes? Can it be used to derive further structural results?

The balanced 2-cut property, as demonstrated in extremal minimal bipartite matching covered graphs, has implications beyond just the extremal classes. This property can be a useful tool in analyzing the connectivity and structure of bipartite graphs in general. By identifying balanced 2-cuts in non-extremal graphs, we can gain insights into the connectivity between different parts of the graph and the distribution of edges. Furthermore, the balanced 2-cut property can be used to derive further structural results by providing a method to decompose graphs into smaller components. This decomposition can help in understanding the relationships between different parts of the graph and identifying key connectivity patterns. Additionally, the presence of balanced 2-cuts can indicate certain symmetries or balanced structures within the graph, which can be leveraged to derive additional structural properties or insights.

Are there any applications or practical implications of the complete characterization of extremal minimal bipartite matching covered graphs?

The complete characterization of extremal minimal bipartite matching covered graphs has several applications and practical implications in various fields such as network analysis, optimization, and graph theory. Some of the key implications include: Network Design: Understanding the structural properties of extremal graphs can help in designing efficient and robust network architectures. By identifying extremal graphs in real-world networks, one can optimize the network layout for better performance and connectivity. Algorithm Development: The insights gained from characterizing extremal graphs can be used to develop algorithms for graph analysis, matching theory, and network optimization. These algorithms can be applied in various domains such as computer science, operations research, and data analytics. Network Resilience: The characterization of extremal graphs can provide insights into the resilience and robustness of networks. By studying extremal graphs, one can identify critical nodes, edges, or structures that are essential for network connectivity and stability. Overall, the complete characterization of extremal minimal bipartite matching covered graphs offers valuable insights that can be applied in network design, algorithm development, and network resilience analysis.
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