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On the Enumeration and Counting of Minimal Dominating sets in Interval and Permutation Graphs [chapter]

Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, Takeaki Uno
2013 Lecture Notes in Computer Science  
We reduce (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumeration of paths in DAGs.  ...  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  ...  In this paper we are interested in the enumeration of (inclusion-wise) minimal dominating sets in interval and permutation graphs.  ... 
doi:10.1007/978-3-642-45030-3_32 fatcat:hx4da5slt5fdjf67osoalev3ty

Minimal Dominating Set Enumeration [chapter]

Mamadou Moustapha Kanté, Lhouari Nourine
2014 Encyclopedia of Algorithms  
If the counting of dominating sets is a #Phard problem and have been investigated in the past [5; 17; 18] , not so much is known for the counting of minimal dominating sets, one can cite few examples:  ...  Another related subject to Dom-Enum is the counting of (minimal) dominating sets in time polynomial in the input graph.  ... 
doi:10.1007/978-3-642-27848-8_721-1 fatcat:52gg7tmopbaujf4b3femw4odra

Page 6181 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
The 4-bases of cardinality 8 with at most three nega- tive elements, and their additive permutations, are enumerated in this paper.  ...  Stout, Perfect dominating sets on cube-connected cycles (51-70); Joseph E. McCanna and Steven J.  ... 

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width [article]

Petr A. Golovach, Pinar Heggernes, Mamadou Moustapha Kanté, Dieter Kratsch, Sigve H. Sæther, Yngve Villanger
2015 arXiv   pre-print
In particular, we show that all 1-minimal and all 1-maximal (\sigma,\rho)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear  ...  Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.  ...  Then, for every colored graph G in one of the following graph classes, we can count in polynomial time, and enumerate with linear delay all 1-minimal (or 1maximal) Red pσ, ρq-dominating sets of G: interval  ... 
arXiv:1509.03753v2 fatcat:ssh2fci7z5fhjb57w5ilnrv3yu

Enumeration of Enumeration Algorithms [article]

Kunihiro Wasa
2016 arXiv   pre-print
In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.  ...  Reference [116] 2.12.7 Enumeration of all minimal dominating sets in an undirected interval graph Input An undirected interval graph G = (V, E). Output All minimal dominating sets.  ...  Reference [117] 2.12.6 Enumeration of all minimal dominating sets in an undirected permutation graph Input An undirected permutation graph G = (V, E). Output All minimal dominating sets.  ... 
arXiv:1605.05102v2 fatcat:bwmhesagkje5hmynqyij7jv224

Page 3075 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
); Giinter Schaar, On “maximal” Hamiltonian cycles in the square of a block (195-198); Jerzy Topp, Graphs with unique minimum edge dominating sets and graphs with unique maximum independent sets of vertices  ...  (199-210); Hoa Vu Dinh [Vi Dinh Hoda], On the length of longest dominating cycles in graphs (211- 222); Bing Wei, Hamiltonian paths and Hamiltonian connectivity in graphs (223-228); Mariusz Wozniak, A  ... 

Common Intervals of Permutations [chapter]

Sylvie Corteel, Guy Louchard, Robin Pemantle
2004 Mathematics and Computer Science III  
Recently several algorithms were designed to efficiently enumerate all common intervals of permutations [9, 17] and their time complexity is O(n + K) where n is the size of the permutation and K the number  ...  An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications.  ...  Acknowledgements Thanks to Kate Davidson (undergraduate research assistant at The University of Pennsylvania) for help with the rigorous numerical bounds in Lemma 4.2 and the computation of the maximum  ... 
doi:10.1007/978-3-0348-7915-6_1 fatcat:qk6dkvmryrg6rmsmjdwcydowjq

Enumerating parametric global minimum cuts by random interleaving

David R. Karger
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
Recently, Aissi et al. gave new counting and algorithmic bounds for parametric minimum cuts in a graph, where each edge cost is a linear combination of multiple cost criteria and different cuts become  ...  minimum as the coefficients of the linear combination are varied.  ...  Aissi et al. give bounds on the number of non-dominated cuts when there are two cost functions, but leave open the question of counting and enumerating such non-dominated cuts in general.  ... 
doi:10.1145/2897518.2897578 dblp:conf/stoc/Karger16 fatcat:qbatawubvreepkvsoiecpyp5xy

There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations ofK14

Petteri Kaski, Patric R. J. Östergård
2009 Journal of combinatorial designs (Print)  
Dieter Kratsch Number of minimal dominating sets. Let ds(n) be the maximum number of minimal dominating sets in a graph on n vertices. It is known that ds(n) ≥ 15 n/6 ≥ 1.5704 n .  ...  Finally, irredundance is the property that makes a dominating set inclusion minimal. Petteri Kaski Counting edge-colorings of the complete graph.  ... 
doi:10.1002/jcd.20188 fatcat:gqreaywcwnacjhodu5oebzzn64

Page 8929 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
Hayer and W. Hochstattler, Test sets for vertex cover problems (5 pp. (electronic)); Arne Hoffmann, Bert Randerath and Lutz Volkmann, Maximal sets of 2-factors in complete equipartite graphs (4 pp.  ...  The paper under review is a survey of some combinatorial prob- lems related to permutations and permutation groups. Special attention is paid to random permutations and the work of Erdos and Turan.  ... 

Belief propagation for permutations, rankings, and partial orders [article]

George T. Cantwell, Cristopher Moore
2022 arXiv   pre-print
We define a continuous spin system whose Gibbs distribution is the posterior distribution on permutations, given a probabilistic model of these interactions.  ...  In addition, the Bethe free energy lets us approximate the number of linear extensions of a partial order and perform model selection between competing probabilistic models, such as the Bradley-Terry-Luce  ...  We thank Jiaming Xu and Jean-Gabriel Young for helpful conversations. Code implementing our methods is available at https://github. com/gcant/pairwise-comparison-BP.  ... 
arXiv:2110.00513v2 fatcat:lq5nkrv3tvg4bjs2c23uueu52m

The fine structure of 321 avoiding permutations [article]

Michael H. Albert
2015 arXiv   pre-print
The notion of absolute indecomposability of a permutation is introduced, and used in enumerating permutations which have a block structure avoiding 321 and whose blocks also have such structure (recursively  ...  Bivariate generating functions for various subsets of the class of permutations containing no descending sequence of length three or more are determined.  ...  Mike Atkinson listened patiently to many preliminary, and incorrect, expositions, and also provided a key step in the proof of Proposition 4. Maple did most of the hard work.  ... 
arXiv:math/0212163v2 fatcat:73a3wylmhbcgda4gmyq4ascqze

Enumerating Minimal Dominating Sets in Triangle-Free Graphs

Marthe Bonamy, Oscar Defrain, Marc Heinrich, Jean-Florent Raymond, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
The objects we wish to enumerate in this paper are the (inclusion-wise) minimal dominating sets of a given graph.  ...  Because dominating sets are among the most studied objects in graph theory and algorithms, their enumeration (and counting) have attracted an increasing attention over the past 10 years.  ...  There is an output-polynomial time algorithm enumerating minimal dominating sets in triangle-free graphs. In particular, the result holds for enumerating minimal dominating sets in bipartite graphs.  ... 
doi:10.4230/lipics.stacs.2019.16 dblp:conf/stacs/BonamyDHR19 fatcat:2qvswqgw3fez5l3fs44c4urql4

Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets [article]

Marthe Bonamy, Oscar Defrain, Piotr Micek, Lhouari Nourine
2020 arXiv   pre-print
It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph.  ...  Since the flipping method is a key tool for the best known algorithms enumerating minimal dominating sets in a number of graph classes, this yields direct improvements on the state of the art.  ...  poset, then there is one enumerating minimal dominating sets in graphs of G.  ... 
arXiv:2004.07214v1 fatcat:5atat6z2hrbzxci2iv7kbumrfa

k-Gap Interval Graphs [chapter]

Fedor V. Fomin, Serge Gaspers, Petr Golovach, Karol Suchan, Stefan Szeider, Erik Jan van Leeuwen, Martin Vatshelle, Yngve Villanger
2012 Lecture Notes in Computer Science  
In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex  ...  Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal.  ...  Corollary 1 . 1 Independent Set, Dominating Set, their weighted and counting versions, and Independent Dominating Set, are FPT on interval+kv graphs.  ... 
doi:10.1007/978-3-642-29344-3_30 fatcat:m7kymxr3hjhhvcyypxfshmqwuy
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