Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs

A hypergraph is a pair pV, Eq where V is a finite set and E Ď 2 is called the set of hyper-edges. An output-polynomial algorithm for C Ď 2 is an algorithm that lists without repetitions all the elements of C in time polynomial in the sum of the size of H and the accumulated size of all the elements in C. Whether there exists an output-polynomial algorithm to list all the inclusion-wise minimal hitting sets of hyper-edges of a given hypergraph (the Trans-Enum problem) is a fifty years old open problem, and up to now there are few tractable examples of hypergraph classes. An inclusion-wise minimal hitting set of the closed neighborhoods of a graph is called a minimal dominating set. A closed neighborhood of a vertex is the set composed of the vertex itself with all its neighbors. It is known that there exists an output-polynomial algorithm for the set of minimal dominating sets in graphs if and only if there is one for the minimal hitting sets in hypergraphs. Hoping this equivalence can help to get new insights in the Trans-Enum problem, it is natural to look at graph classes. It was proved independently and with different techniques in [Golovach et al. ICALP 2013] and [Kanté et al. ISAAC 2012] that there exists an incremental output-polynomial algorithm for the set of minimal edge dominating sets in graphs (i.e. minimal dominating sets in line graphs). We provide the first polynomial delay and polynomial space algorithm that lists all the minimal edge dominating sets in graphs, answering an open problem of [Golovach et al. ICALP 2013]. Besides the result, we hope the used techniques that are a mix of a modification of the well-known Berge’s algorithm and a strong use of the structure of line graphs, are of great interest and could be used to get new output-polynomial algorithms.