Efficient dominating sets in Cayley graphs

An independent set C of vertices in a graph is an efficient dominating set (or perfect code) when each vertex not in C is adjacent to exactly one vertex in C. An E-chain is a countable family of nested graphs, each of which has an efficient dominating set. The Hamming codes in the n-cubes provide a classical example of E-chains. We give a constructing tool to produce E-chains of Cayley graphs. This tool is used to construct infinite families of E-chains of Cayley graphs on symmetric groups. These families include the well-known star graphs, for which the efficient domination property was proved by Arumugam and Kala, and pancake graphs. Additional structural properties of the E-chains and the efficient dominating sets involved are also presented. Given a tree T , the T -graph associated to T seems to be a natural candidate of a graph with an efficient dominating set. However, we prove that a T -graph has an efficient dominating set if and only if T is a star.