Graph Domination and Minimum Rank Theory
Summary
Graph domination encompasses a family of invariants measuring the efficiency with which selected vertices control or monitor the remainder of a network. The classical domination number quantifies the minimum size of a vertex set such that every node is either in the set or adjacent to a chosen dominator. Extensions such as power domination incorporate propagation rules inspired by electrical‐grid observability, while k-fault-tolerant variants model resilience against component failures. In parallel, minimum rank theory seeks the smallest possible rank among real symmetric matrices whose zero–nonzero patterns reflect a graph’s edges. Central to this connection is the zero forcing number, which bounds the maximum nullity of associated matrices and thus governs minimum rank. These intertwined frameworks underpin applications in network security, power-grid monitoring, quantum control and combinatorial optimisation, even as many related decision problems remain computationally intractable.
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Graph Domination and Minimum Rank Theory publication trend
The graph below shows the total number of articles in graph domination and minimum rank theory across all publications each year (not limited to Nature Index journals).
Technical terms
Dominating set: A subset of vertices such that every vertex in the graph is either in the subset or adjacent to at least one vertex in the subset; the domination number is the minimum cardinality of such a set.
Power dominating set: A vertex set that, under rules modelling electrical monitoring, can observe all vertices and edges via propagation; the power domination number is its minimum cardinality.
k-fault-tolerant power dominating set: A power dominating set that remains valid after the removal of any up to k of its vertices; the k-fault-tolerant power domination number is the smallest size of such a resilient set.
Zero forcing set: An initial colouring of vertices from which all vertices become coloured by iteratively forcing uniquely adjacent uncoloured neighbours; the zero forcing number is the minimum size of such a set.
Minimum rank of a graph: The smallest rank among all real symmetric matrices whose pattern of nonzero off-diagonal entries corresponds to the edges of the graph; closely related to maximum nullity as vertex count minus minimum rank.
References
- Bound for the k-Fault-Tolerant Power-Domination Number. Symmetry (2024).
- Characterization of All Graphs with a Failed Skew Zero Forcing Number of 1. Mathematics (2022).
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