Domination Theory in Graph Structures

Domination theory explores subsets of vertices in a graph such that every vertex lies in or adjacent to this subset. The smallest such subset defines the domination number, a central invariant that encodes coverage and control in networked systems. Over recent decades, generalisations have proliferated to capture constraints in monitoring, resource allocation and resilience. These include total domination, where every vertex must be adjacent to a dominator; Roman domination, inspired by defence strategies with multiple levels of support; and weighted or Italian domination variants to model capacities or multifaceted coverage. The study encompasses existence conditions, extremal bounds, algorithmic complexity and constructive methods, spanning classical combinatorial techniques and modern heuristic and exact algorithms. Domination parameters find applications in sensor placement, facility location, social influence and bio-network analysis, with real-world graphs driving advances in both theoretical limits and scalable computation.

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Recent advances have addressed both new domination variants and algorithmic frameworks. A novel k-weighted dominating set problem extends classical domination to weighted graphs, defining the k-weighted domination number as the minimum vertex subset whose adjacent weight sum meets a threshold; heuristic and iterated greedy algorithms demonstrate near-optimal performance within significantly reduced computational time. In the domain of Roman domination, surveys of multiple varieties of Roman dominating functions—including signed and constrained forms—provide a unified treatment of definitions and known bounds, identifying key relationships between parameters and characterising extremal graph families. For product graphs, ensemble and partitioning methods have been applied to the Italian domination number on Cartesian products of cycles and paths, yielding exact values for certain grid dimensions and establishing general bounds for larger instances. These works highlight the interplay of combinatorial bounds, complexity analysis and tailored algorithms for specialised graph classes, paving the way for handling large-scale and diverse real-world networks.

Domination Theory in Graph Structures publication trend

The graph below shows the total number of articles in domination theory in graph structures 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 this subset or adjacent to at least one member of it.

Domination number (γ): The minimum cardinality of a dominating set in a graph.

k-weighted dominating set: A generalisation where each edge has a weight and each dominated vertex must receive a total weight of at least k from adjacent chosen vertices; the k-weighted domination number is the size of the smallest such set.

Roman dominating function: A labelling of vertices with 0, 1 or 2 such that each vertex labelled 0 is adjacent to at least one vertex labelled 2; the Roman domination number is the minimum total label sum.

Italian dominating function: A function from vertices to {0,1,2} such that each vertex labelled 0 has adjacent labels summing to at least 2; the Italian domination number is the minimum sum of labels.

References

1. Finding the minimum k -weighted dominating sets using heuristic algorithms. Mathematics and Computers in Simulation (2025).
2. Varieties of Roman domination II. AKCE International Journal of Graphs and Combinatorics (2020).
3. Bagging Approach for Italian Domination in $C_{n} \square\,P_{m}$. IEEE Access (2019).