A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γ c (G) is the minimum size of such a set. Let $${\delta^*(G)={\rm min}\{\delta(G),\delta({\overline{G}})\}}$$, where $${{\overline{G}}}$$ is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and $${{\overline{G}}}$$ are both connected, $${{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+4-({\gamma_c}(G)-3)({\gamma_c}({\overline{G}})-3)}$$. As a corollary, $${{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \frac{3n}{4}}$$ when δ*(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that $${{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+2}$$ when $${{\gamma_c}(G),{\gamma_c}({\overline{G}})\ge 4}$$. This bound is sharp when δ*(G) = 6, and equality can only hold when δ*(G) = 6. Finally, we prove that $${{\gamma_c}(G){\gamma_c}({\overline{G}})\le 2n-4}$$ when n ≥ 7, with equality only for paths and cycles.
['Hosein Karami', 'Seyed Mahmoud Sheikholeslami', 'Abdollah Khodkar', 'Douglas B. West']
Connected Domination Number of a Graph and its Complement