On the maximum number of minimal connected dominating sets in convex bipartite graphs

The enumeration of minimal connected dominating sets is known to be notoriously hard for general graphs. Currently, it is only known that the sets can be enumerated slightly faster than $\mathcal{O}^{*}(2^n)$ and the algorithm is highly nontrivial. Moreover, it seems that it is hard to use bipartiteness as a structural aide when constructing enumeration algorithms. Hence, to the best of our knowledge, there is no known input-sensitive algorithm for enumerating minimal dominating sets, or one of their related sets, in bipartite graphs better than that of general graphs. In this paper, we provide the first input-sensitive enumeration algorithm for some non trivial subclass of bipartite graphs, namely the convex graphs. We present an algorithm to enumerate all minimal connected dominating sets of convex bipartite graphs in time $\mathcal{O}(1.7254^{n})$ where $n$ is the number of vertices of the input graph. Our algorithm implies a corresponding upper bound for the number of minimal connected dominating sets for this graph class. We complement the result by providing a convex bipartite graph, which have at least $3^{(n-2)/3}$ minimal connected dominating sets.