Domination in 4-Regular Graphs with Girth 3

Abstract

In this paper, we investigate the domination number, independent domination number, connected domination number, total domination number denoted by \( \gamma (G\left( n \right)), \gamma_{i} (G\left( n \right)), \gamma_{c} (G\left( n \right),\gamma_{t} (G\left( n \right)) \) respectively for 4-regular graphs of n vertices with girth 3. Here, G(n) denotes the 4-regular graphs of n vertices with girth 3. We obtain some exact values of G(n) for these parameters. We further establish that \( \gamma_{i} \left( {G\left( n \right)} \right) = \gamma \left( {G\left( n \right)} \right)\, {\text{for }} n \ge 6 \) and \( \gamma_{c} \left(G\left( n \right)\right) = \gamma_{t} \left( {G\left( n \right)} \right) \) for n ≥ 6. Nordhaus–Gaddum type results are also obtained for these parameters.