Connected dominating set in graphs without long paths and cycles

The ratio of the connected domination number, γc, and the domination number, γ, is strictly bounded from above by 3. It was shown by Zverovich that for every connected (P5, C5)-free graph, γc = γ. In this paper, we investigate the interdependence of γ and γc in the class of (Pk, Ck)-free graphs, for k ≥ 6. We prove that for every connected (P6, C6)-free graph, γc ≤ γ+1 holds, and there is a family of (P6, C6)-free graphs with arbitrarily large values of γ attaining this bound. Moreover, for every connected (P8, C8)-free graph, γc/γ ≤ 2, and there is a family of (P7, C7)-free graphs with arbitrarily large values of γ attaining this bound. In the class of (P9, C9)-free graphs, the general bound γc/γ ≤ 3 is asymptotically sharp. keywords: domination, connected domination, forbidden induced subgraphs. MSC: 05C69, 05C75, 05C38.