Journal of Combinatorial Theory Series B
A Group-Theoretic Model for Symmetric Interconnection Networks
IEEE Transactions on Computers
Graphs and Hypergraphs
Efficient domination in circulant graphs with two chord lengths
Information Processing Letters
Reinforcement numbers of digraphs
Discrete Applied Mathematics
On perfect codes in Cartesian products of graphs
European Journal of Combinatorics
Automorphism groups of the Pancake graphs
Information Processing Letters
Efficient domination in cubic vertex-transitive graphs
European Journal of Combinatorics
On some structural properties of star and pancake graphs
Information Theory, Combinatorics, and Search Theory
Discrete Applied Mathematics
Total and paired domination numbers of toroidal meshes
Journal of Combinatorial Optimization
Designs, Codes and Cryptography
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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.