Domination and total domination on asteroidal triple-free graphs

We present the first polynomial time algorithms for solving the NP-complete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triple-free graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating set on a large superclass of the asteroidal triple-free graphs, namely the class of those graphs for which each connected component has a so-called dominating shortest path. Our algorithms can be implemented to run in time O(n⁶) on asteroidal-triple free graphs and in time O(n⁷) on graphs with dominating shortest path.