1 research outputs found
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 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 where 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
minimal connected dominating sets.Comment: 10 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1602.07504 by other author