On the Enumeration of Minimal Dominating Sets and Related Notions
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine
SIAM Journal on Discrete Mathematics
A dominating set D in a graph is a subset of its vertex set such that each vertex is either in D or has a neighbour in D. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be
... y reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in P 6 -free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs. M.M. Kanté and V. A preliminary version of Sections 4 and 5 appeared in the proceedings of FCT 2011. 1 arXiv:1407.2053v1 [cs.DM] 8 Jul 2014 minimal dominating sets of a graph is in bijection with the set of minimal transversals of its closed neighbourhood hypergraph  . The Trans-Enum problem has been intensively studied due to its connections to several problems in such fields as data-mining and learning [11, 12, 16, 20, 24] . It is still open whether there exists an output-polynomial time algorithm for the Trans-Enum problem, but several classes where an output-polynomial time algorithm exists have been identified (see for instance the survey ). So, classes of graphs whose closed neighbourhood hypergraphs are in one of these identified classes of hypergraphs admit also outputpolynomial time algorithms for the Dom-Enum problem. Examples of such graph classes are planar graphs and bounded degree graphs (see [18, 19] for more information). Recently, the Dom-Enum problem has been studied by several groups of authors [8, 14] . Their research on exact exponential-time algorithms triggered a new approach to the design of enumeration algorithms which uses classical worstcase running time analysis, i.e., the running time depends on the length of the input. In this paper, we first prove that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. Since the other direction also holds, the two problems are equivalent, i.e., there exists an output-polynomial time algorithm for the Dom-Enum problem if and only if there exists one for the Trans-Enum problem. One could possibly expect to benefit from graph theory tools to solve the two problems and at the same time many other enumeration problems equivalent to the Trans-Enum problem (see  for examples of problems equivalent to Trans-Enum). In addition, we show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: TDom-Enum problem, CDom-Enum in split graphs, TDom-Enum in split graphs, Dom-Enum in cobipartite graphs, where the TDom-Enum problem corresponds to the enumeration of minimal total dominating sets. We then characterise graphs where the addition of edges changes the set of minimal dominating sets. The maximal extension (addition of edges) that keeps invariant the set of minimal dominating sets can be computed in polynomial time, and appears to be a useful tool for getting output-polynomial time algorithms for the Dom-Enum problem in new graph classes such as P 6 -free chordal graphs. As a consequence, Dom-Enum in split graphs and Dom-Enum in P 6 -free chordal graphs are linear delay and polynomial space. We finally study the complexity of the enumeration of minimal connected dominating sets (called the CDom-Enum problem). The Minimum Connected Dominating Set problem is a well-known and well-studied variant of the Minimum Dominating Set problem due to its applications in networks [17, 28] . We have proved in  that CDom-Enum in split graphs is equivalent to the Trans-Enum problem. We will extend this result to other graph classes. Indeed, we prove that the minimal connected dominating sets of a graph are the minimal transversals of its minimal separators. As a consequence, in any class of graphs with a polynomially bounded number of minimal separators, the CDom-Enum problem can be polynomially reduced to the Trans-Enum problem; examples of such classes are chordal graphs, circle graphs and circular-arc graphs [5, 21, 23] . Finally, we show that the CDom-Enum problem is harder than the Dom-Enum problem.