Enumerating Connected Dominating Sets

Abstract

The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2 n is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time O(1.9896 n ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time O(1.9767 n ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890 n ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422 n ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195 n ) and Ω(1.4723 n ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U , if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U |. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.