On generating all maximal independent sets
Information Processing Letters
Exact transversal hypergraphs and application to Boolean &mgr;-functions
Journal of Symbolic Computation
Identifying the Minimal Transversals of a Hypergraph and Related Problems
SIAM Journal on Computing
Reverse search for enumeration
Discrete Applied Mathematics - Special volume: first international colloquium on graphs and optimization (GOI), 1992
On the complexity of dualization of monotone disjunctive normal forms
Journal of Algorithms
Polynomial-Time Recognition of 2-Monotonic Positive Boolean Functions Given by an Oracle
SIAM Journal on Computing
The Maximum Latency and Identification of Positive Boolean Functions
SIAM Journal on Computing
A fast and simple algorithm for identifying 2-monotonic positive Boolean functions
Journal of Algorithms
An Algorithm to Enumerate All Cutsets of a Graph in Linear Time per Cutset
Journal of the ACM (JACM)
New Results on Monotone Dualization and Generating Hypergraph Transversals
SIAM Journal on Computing
NP-Completeness: A Retrospective
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Hypergraph Transversal Computation and Related Problems in Logic and AI
JELIA '02 Proceedings of the European Conference on Logics in Artificial Intelligence
Generating All Vertices of a Polyhedron Is Hard
Discrete & Computational Geometry
Linear delay enumeration and monadic second-order logic
Discrete Applied Mathematics
Enumeration of minimal dominating sets and variants
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
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We show that all minimal edge dominating sets of a graph can be generated in incremental polynomial time. We present an algorithm that solves the equivalent problem of enumerating minimal (vertex) dominating sets of line graphs in incremental polynomial, and consequently output polynomial, time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output polynomial time is a fundamental and challenging question; it has been open for several decades and has triggered extensive research. To obtain our result, we present a flipping method to generate all minimal dominating sets of a graph. Its basic idea is to apply a flipping operation to a minimal dominating set D* to generate minimal dominating sets D such that G[D] contains more edges than G[D*]. We show that the flipping method works efficiently on line graphs, resulting in an algorithm with delay $O(n^2m^2|\mathcal{L}|)$ between each pair of consecutively output minimal dominating sets, where n and m are the numbers of vertices and edges of the input graph, respectively, and $\mathcal{L}$ is the set of already generated minimal dominating sets. Furthermore, we are able to improve the delay to $O(n^2 m|\mathcal{L}|)$ on line graphs of bipartite graphs. Finally we show that the flipping method is also efficient on graphs of large girth, resulting in an incremental polynomial time algorithm to enumerate the minimal dominating sets of graphs of girth at least 7.