On the Enumeration and Counting of Minimal Dominating sets in Interval and Permutation Graphs
نویسندگان
چکیده
We reduce (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumeration of paths in DAGs. As a consequence, we can enumerate in linear delay, after a polynomial time pre-processing, minimal dominating sets in interval and permutation graphs. We can also count them in polynomial time. This improves considerably upon previously known results on interval graphs, and to our knowledge no output polynomial time algorithm for the enumeration of minimal dominating sets and their counting were known for permutation graphs.
منابع مشابه
On the Enumeration of Minimal Dominating Sets and Related Notions
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 an output-sensitive enumeration algorithm of (inclusionwise) minimal dominating sets in graphs, called Dom problem. It was known that this problem can be polynomially reduced to the well known Transversal problem in hypergraphs. We show th...
متن کاملEnumeration of Minimal Dominating Sets and Variants
In this paper, we are interested in the enumeration of minimal dominating sets in graphs. A polynomial delay algorithm with polynomial space in split graphs is presented. We then introduce a notion of maximal extension (a set of edges added to the graph) that keeps invariant the set of minimal dominating sets, and show that graphs with extensions as split graphs are exactly the ones having chor...
متن کاملStrength of strongest dominating sets in fuzzy graphs
A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...
متن کاملGenerating All Minimal Edge Dominating Sets with Incremental-Polynomial Delay
For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m|L|) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m|L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set wi...
متن کاملOn the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets
We prove that line graphs and path graphs have bounded neighbourhood Helly. As a consequence, we obtain output-polynomial time algorithms for enumerating the set of minimal dominating sets of line graphs and path graphs. Therefore, there exists an output-polynomial time algorithm that enumerates the set of minimal edge-dominating sets of any graph.
متن کامل