turkmath.org

A dominating set of $\Gamma$ is a subset $D$ of $V$, such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number $\gamma(\Gamma)$ is the minimum cardinality of a dominating set for $\Gamma$. An efficient dominating set of $\Gamma$ is an independent subset $D$ of $V$, such that every vertex not in $D$ is adjacent to exactly one member of $D$. Observe that if $D$ is an efficient dominating set of $\Gamma$, then $D$ is also minimum dominating set of $\Gamma$.

There are results on the domination number and finding efficient dominating sets (if exist) on circulant graphs for certain connection sets. Here I will talk about our results on odd-regular circulant graphs and Cayley graphs on different groups by giving the exact domination number when possible or giving a meaningful bound.