Connected domatic number - tlooto, The Most Powerful AcademicGPT

The connected domatic number of a graph, denoted $$dc(G)$$, is a fundamental concept in graph theory that extends the classical notions of domination and domatic partition by also demanding connectivity within dominating sets. Here, we synthesize and expand upon the provided materials, addressing both theoretical underpinnings and practical implications. --- Formal Definition In a connected graph $$G = (V, E)$$, a connected dominating set (CDS) is a subset $$D \subseteq V$$ such that: - Every vertex $$v \in V \setminus D$$ is adjacent to at least one vertex in $$D$$ (domination), and - The subgraph induced by $$D$$, $$G[D]$$, is connected. A connected domatic partition is a partition of $$V$$ into pairwise disjoint CDSs. The connected domatic number $$dc(G)$$ is the largest integer $$k$$ such that $$V$$ can be partitioned into $$k$$ disjoint connected dominating sets[1][2]. Mathematically, $$ dc(G) = \max \{k : V = D1 \cup D2 \cup \cdots \cup Dk, \; Di \cap Dj = \emptyset \text{ for }