Matching Preclusion in Graph Theory
Summary
Matching preclusion examines the resilience of a network modelled as a graph by identifying the smallest set of edge removals that destroys all perfect matchings. Originating in the study of interconnection networks, this invariant quantifies fault tolerance by pinpointing minimal “obstruction” sets whose deletion precludes the existence of a perfect pairing of vertices. Over the past decade, the concept has been enriched by variations such as conditional matching preclusion, which requires the post-deletion graph to remain free of isolated vertices, and fractional matching preclusion, which generalises the notion to weighted or capacity-constrained edges. Applications span processor-communication frameworks, where uninterrupted data exchange relies on alternative matching routes, to molecular chemistry, where perfect matchings model stable bonding configurations. The metric has been determined exactly for numerous families, including hypercubes, complete bipartite graphs and various hierarchical or derived network topologies, offering both theoretical insight and practical guidelines for designing robust architectures.
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Recent advances have sharpened bounds and characterised extremal cases for matching-preclusion variants. A 2023 study on conditional matching preclusion established tight upper and lower bounds for the conditional matching preclusion number in general graphs, classified those with extreme parameter values and resolved related extremal problems. In 2022, analytic techniques were applied to hierarchical cubic networks to determine both their matching preclusion and conditional matching preclusion numbers, alongside a complete description of all minimal edge sets whose removal forces the network to lose perfect or almost-perfect matchings. Earlier foundational work extended the framework to bipartite structures: by defining a hierarchy of matching preclusion properties, researchers characterised complete bipartite graphs and hypercubes in terms of their resistance to edge failures under increasingly stringent non-isolation constraints. These contributions collectively deepen our understanding of network robustness, revealing how structural symmetries and connectivity patterns influence the ease with which perfect matchings can be disrupted.
Matching Preclusion in Graph Theory publication trend
The graph below shows the total number of articles in matching preclusion in graph theory across all publications each year (not limited to Nature Index journals).
Technical terms
Graph: A collection of vertices joined pairwise by edges, modelling relational structures.
Perfect matching: A set of edges without common vertices that covers every vertex in the graph.
Matching preclusion number: The minimum number of edges whose deletion results in a graph devoid of any perfect matching.
Conditional matching preclusion number: The minimum number of edges whose deletion yields a graph with no isolated vertices and no perfect matching.
References
- Conditional Matching Preclusion Number of Graphs. Discrete Dynamics in Nature and Society (2023).
- Matching preclusion and conditional matching preclusion for hierarchical cubic networks. AIMS Mathematics (2022).
- Fractional matching preclusion for butterfly derived networks. Theory and Applications of Graphs (2019).
- Generalized Matching Preclusion in Bipartite Graphs. Theory and Applications of Graphs (2018).
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