Efficient dominating sets in circulant graphs

Let S be a subset of finite cyclic group Zn not containing the identity element 0 with S=−S. Cayley graphs on Zn with respect to S are called circulant graphs and denoted by Cay(Zn,S). In this paper, for connected non-complete circulant graphs Cay(Zn,S) of degree |S|=p−1 with p prime, we give a necessary and sufficient condition for the existence of efficient dominating sets, and characterize all efficient dominating sets if exist. We also obtain similar results for Cay(Zn,S) of degree |S|=pq−1 and pm−1, where p,q are primes, m is a positive integer, and |S|+1 is relatively prime to n|S|+1. Moreover, we give a necessary and sufficient condition for the existence of efficient dominating sets in Cay(Zn,S) of order n=pkq,p2q2,pqr,p2qr,pqrs and degree |S|, where p,q,r,s are distinct primes, k is a positive integer, and |S|+1 is relatively prime to n|S|+1.