568 research outputs found
Enumerating Minimal Dominating Sets in Triangle-Free Graphs
International audienceIt is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem
Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality
Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes
Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs
In [M. M. Kant\'e, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of
minimal dominating sets and related notions. SIAM Journal on Discrete
Mathematics, 28(4):1916-1929, 2014] the authors give an delay
algorithm based on neighborhood inclusions for the enumeration of minimal
dominating sets in split and -free chordal graphs. In this paper, we
investigate generalizations of this technique to -free chordal graphs for
larger integers . In particular, we give and delays
algorithms in the classes of -free and -free chordal graphs. As for
-free chordal graphs for , we give evidence that such a technique
is inefficient as a key step of the algorithm, namely the irredundant extension
problem, becomes NP-complete.Comment: 16 pages, 3 figure
Counting dominating sets and related structures in graphs
We consider some problems concerning the maximum number of (strong)
dominating sets in a regular graph, and their weighted analogues. Our primary
tool is Shearer's entropy lemma. These techniques extend to a reasonably broad
class of graph parameters enumerating vertex colorings satisfying conditions on
the multiset of colors appearing in (closed) neighborhoods. We also generalize
further to enumeration problems for what we call existence homomorphisms. Here
our results are substantially less complete, though we do solve some natural
problems
- …