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### Independent Domination in Directed Graphs [article]

Michael Cary, Jonathan Cary, Savari Prabhu
2019 arXiv   pre-print
We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general.  ...  We also provide the idomatic number for special cases of some of these families of digraphs.  ...  Trees We next turn our attention to orientations of trees by proving an existence theorem for all orientations of trees. Theorem 4. Every oriented tree admits an independent dominating set, Proof.  ...

### Page 7332 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews
We also examine the maximum number y, of orientations admitting an efficient dominating set in a tree with g edges.  ...  , Anthony E. (1-WILC; La Crosse, W1); Clark, Lane H. (1-SIL; Carbondale, IL) The number of orientations of a tree admitting an efficient dominating set.  ...

### Page 2313 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews
The set D is a total dominating set in G if every vertex of G is adjacent to some vertex of D, and D is an independent dominating set in G if D is a dominating set and D is also independent.  ...  The minimum orders of dominating sets, total dominating sets, and independent dominating sets, denoted 7(G), +:(G) and 7(G), respectively, are called the domination number, the total domination number  ...

### Efficient total domination in digraphs

Oliver Schaudt
2012 Journal of Discrete Algorithms
An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X.  ...  We study graphs that permit an orientation having such a set and give complexity results and characterizations.  ...  Acknowledgement I am indebted to an anonymous referee, whose careful suggestions helped to improve the presentation of the paper a lot.  ...

### Title

Robert Ganian, Thekla Hamm, Topi Talvitie
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
The problem has been studied, among others, in the context of causal discovery, and it is known that it reduces to counting the number of so-called moral acyclic orientations of certain undirected graphs  ...  We consider the problem of counting the number of DAGs which are Markov-equivalent, i.e., which encode the same conditional independencies between random variables.  ...  Robert Ganian and Thekla Hamm acknowledge support from the Austrian Science Fund (FWF, Project P31336: NFPC). Thekla Hamm is co-funded by FWF project W1255-N23.  ...

### Forthcoming Papers

1990 Discrete Mathematics
(such as being a minimum dominating set) or finding a partition of the vertex set into such sets (for example, a partition into the maximum possible number of dominating sets).  ...  We study a new version of the domination problem in which the dominating set is required to be a clique.  ...  The least domination number y,(G) of G is the minimum cardinality of an 1.d.s. If v, is the total domination number, we prove among other things: 0) v1 s v,, (ii) for a tree, vr s or.  ...

### Plane Geometric Graph Augmentation: A Generic Perspective [chapter]

Ferran Hurtado, Csaba D. Tóth
2012 Thirty Essays on Geometric Graph Theory
The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known.  ...  We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane.  ...  They also show that every multigraph that has an even number of edges and contains two edge-disjoint spanning trees must have a conflict-free even orientation.  ...

### Page 4741 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews
Summary: “A graph G has tree-width at most w if it admits a tree-decomposition of width < w.  ...  Summary: “Efficient algorithms are developed for finding a min- imum cardinality connected dominating set and a minimum car- dinality Steiner tree in permutation graphs.  ...

### Counting and Sampling Markov Equivalent Directed Acyclic Graphs

Topi Talvitie, Mikko Koivisto
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms.  ...  2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n.  ...  Acknowledgments This work was supported in part by the Academy of Finland, under Grant 276864.  ...

### Fast algorithms for finding Hamiltonian paths and cycles in in-tournament digraphs

Jørgen Bang-Jensen, Pavol Hell
1993 Discrete Applied Mathematics
An in-tournament digraph is a digraph in which the set of in-neighbours of every vertex induces a tournament.  ...  Here n, m denote the number of vertices respectively arcs of the in-tournament digraph.  ...  A local tournament digraph (local tournament for short) [2, 51 , is an oriented graph in which the set of in-neighbours as well as the set of out-neighbours of each vertex induce a tournament.  ...

### Locating-dominating sets: from graphs to oriented graphs [article]

Nicolas Bousquet, Quentin Deschamps, Tuomo Lehtilä, Aline Parreau
2022 arXiv   pre-print
A locating-dominating set in an oriented graph is a set S such that for every w∈ V, N[w]^-∩ S=∅ and for each pair of vertices u,v∈ V∖ S, N^-(u)∩ S N^-(v)∩ S. We consider the following two parameters.  ...  Given an undirected graph G, we look for →γ_LD(G) (→Γ_LD(G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G.  ...  Moreover, if we take a cyclic orientation of C n , the set of vertices with an odd index number forms an optimal locating-dominating set.  ...

### Algorithmic Properties of Sparse Digraphs [article]

Stephan Kreutzer, Patrice Ossona de Mendez, Roman Rabinovich, Sebastian Siebertz
2017 arXiv   pre-print
More specifically, we show that the directed Steiner tree problem is fixed-parameter tractable on any class of directed bounded expansion parameterized by the number k of non-terminals plus the maximal  ...  up to a factor O(k· k) on any class of directed bounded expansion, where k denotes the size of an optimal solution.  ...  Most generally, it admits an O(log k) approximation on graphs of bounded VC-dimension  (where k is the size of an optimal dominating set).  ...

### Recognition and Orientation Algorithms for P4-Comparability Graphs [chapter]

Stavros D. Nikolopoulos, Leonidas Palios
2001 Lecture Notes in Computer Science
P4-transitive orientation of a P4-comparability graph.  ...  We consider two problems pertaining to P4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P4-comparability graph and the problem of producing an acyclic  ...  A graph which admits an acyclic transitive orientation is called a comparability graph [3, 4] ; A graph is a P 4 -comparability graph if it admits an acyclic P 4 -transitive orientation [5, 6] .  ...

### Survey of local algorithms

Jukka Suomela
2013 ACM Computing Surveys
A local algorithm is a distributed algorithm that runs in constant time, independently of the size of the network.  ...  Encoding of input and output. When we study local algorithms, we assume that the problem instance is given in a distributed manner: each node in the communication graph G knows part of the input.  ...  A preliminary version of this survey was published as a part of my PhD thesis [134, §2]. Updates to this work will be posted online at http://www.  ...

### Splitting trees

Pierre Hansen, Alain Hertz, Nicolas Quinodoz
1997 Discrete Mathematics
An O(N) algorithm is proposed to get an upper bound c'(t), the connected splitting number, on the splitting number o(t) of a tree T and an O(N log N) algorithm to compute this last number, where N is the  ...  number of vertices of the tree.  ...  Acknowledgements Work of the first author was done during a visit to ficole Polytechnique Ftdtrale de Lausanne, whose hospitality is gratefully acknowledged.  ...
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