Domatic number¶
Assign a graph the largest number of blocks in a vertex partition for which every block is a dominating set.
Core Idea¶
The domatic number \(d(G)\) is the maximum integer \(k\) for which \(V(G)\) has a partition \(\{D_1,\ldots,D_k\}\) and every block \(D_i\) is a dominating set of \(G\). Each block must meet the closed neighborhood of every vertex, while partition disjointness makes blocks compete for the vertices available in small closed neighborhoods; maximizing the number of such simultaneously dominating blocks turns a feasibility family into a graph invariant.
Its autonomous residual is the maximum cardinality of a vertex partition whose every block independently dominates the whole graph, not merely a minimum dominating-set size or any partition into graph-theoretic special sets.
Scope of Application¶
Domatic number applies when the analyst can specify a finite undirected graph \(G=(V,E)\), its closed neighborhoods, and partitions of the vertex set \(V\) and establish that the blocks are nonempty, pairwise disjoint, collectively exhaustive over \(V(G)\), each block intersects every closed neighborhood, and no partition satisfying those constraints has more blocks. The entry states a mathematical invariant and descriptive complexity results, not an operational network-deployment recipe. Variant names, graph conventions, and complexity assumptions must remain explicit.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because domatic may be mistaken for domination generally, while number can refer to the optimum value, the associated decision problem, or a variant such as total domatic number. The disciplined statement is that the object counts as Domatic number exactly when the blocks are nonempty, pairwise disjoint, collectively exhaustive over \(V(G)\), each block intersects every closed neighborhood, and no partition satisfying those constraints has more blocks
Manages Complexity¶
The abstraction compresses complete and bipartite graphs, regular graphs, graphs with isolated vertices, total and connected variants, weighted and fractional generalizations, exact algorithms, approximations, and fixed graph classes into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Abstract Reasoning¶
- Type the carrier. Establish a finite undirected graph \(G=(V,E)\), its closed neighborhoods, and partitions of the vertex set \(V\) and reject examples from a different problem. 2. Lock the rule. Express that the blocks are nonempty, pairwise disjoint, collectively exhaustive over \(V(G)\), each block intersects every closed neighborhood, and no partition satisfying those constraints has more blocks independently of one notation or implementation.
Knowledge Transfer¶
Transfer within graph theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For a complete graph \(K_n\), the \(n\) singleton vertex blocks each dominate the graph, so \(d(K_n)=n\). to If \(G\) has minimum degree \(\delta\), then \(d(G)\leq\delta+1\), because every dominating block must contain a vertex from the closed neighborhood of a minimum-degree vertex. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Domatic number Domain-specific
Parents (1) — more general patterns this builds on
-
Domatic number is a kind of Partition Prime
The proposed strict upward parent is
prime:partition.
Hierarchy path (1) — routes to 1 parentless root
- Domatic number → Partition → Set and Membership
Neighborhood in Abstraction Space¶
Domatic number sits in a crowded region of the domain-specific corpus (28th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Connectivity & Network Measures (31 abstractions)
Nearest neighbors
- Component (graph theory) — 0.91
- Shortest path problem — 0.91
- Deficiency (graph theory) — 0.91
- Strength of a graph — 0.91
- Split graph — 0.90
Computed from structural-signature embeddings · 2026-09-08