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### Improved Algorithms and Complexity Results for Power Domination in Graphs

Jiong Guo, Rolf Niedermeier, Daniel Raible
2007 Algorithmica
The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P ⊆ V such that all vertices in V are "observed" by vertices  ...  Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs.  ...  For a vertex v in graph G, we denote by N G (v) the open neighborhood of v in G. By N G [v] , we refer to the closed neighborhood of v.  ...

### Improved Algorithms and Complexity Results for Power Domination in Graphs [chapter]

Jiong Guo, Rolf Niedermeier, Daniel Raible
2005 Lecture Notes in Computer Science
The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P ⊆ V such that all vertices in V are "observed" by vertices  ...  Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs.  ...  For a vertex v in graph G, we denote by N G (v) the open neighborhood of v in G. By N G [v] , we refer to the closed neighborhood of v.  ...

### Fatgraph Algorithms and the Homology of the Kontsevich Complex [article]

Riccardo Murri
2012 arXiv   pre-print
This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms of any given fatgraph; (3) compute the homology of  ...  the fatgraph complex.  ...  (and particularly the "Theoretical Chemistry and Computational Grid Applications" research group led by Kim Baldridge), whose support is here gratefully acknowledged.  ...

### Partial Domination in Graphs [article]

Benjamin M. Case, Stephen T. Hedetniemi, Renu C. Laskar, Drew J. Lipman
2017 arXiv   pre-print
In this paper we propose relaxing this requirement, by seeking sets of vertices that dominate a prescribed fraction of the vertices of a graph.  ...  A set S⊆ V is a dominating set of G if every vertex in V - S is adjacent to at least one vertex in S.  ...  We conclude our look at computational complexity by noting an interesting open question. Open Question 2.  ...

### RAINBOW DOMINATION IN GRAPHS

Boštjan Brešar, Michael A. Henning, Douglas F. Rall
2008 Taiwanese journal of mathematics
If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this is called the k-rainbow dominating function of a graph G.  ...  In this paper we connect this new concept to usual domination in (products of) graphs, and present its application to paired-domination of Cartesian products of graphs.  ...  For any vertex v ∈ V , the open neighborhood of v is the set N (v) = {u ∈ V | uv ∈ E}, and its closed neighborhood is the set N [v] = N (v) ∪ {v}.  ...

### Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments

Peter Hachenberger, Lutz Kettner, Kurt Mehlhorn
2007 Computational geometry
In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary  ...  The implementation was released as Open Source in the Computational Geometry Algorithm Library (CGAL) release 3.1 in December 2004.  ...  and Surfaces).  ...

### Conflict-Free Coloring of Graphs

Zachary Abel, Victor Alvarez, Erik D. Demaine, Sándor P. Fekete, Aman Gour, Adam Hesterberg, Phillip Keldenich, Christian Scheffer
2018 SIAM Journal on Discrete Mathematics
We thank Bruno Crepaldi, Pedro de Rezende, Cid de Souza, Stephan Friedrichs, Michael Hemmer, and Frank Quedenfeld for helpful discussions.  ...  Moreover, Corollary 5.1 discusses the complexity of k-Conflict-Free Dominating Set, and Theorem 5.2 states positive results for outerplanar graphs. Theorem 5.2.  ...  Kikuno, Yoshida, and Kakuda [23] present a linear time algorithm for finding a minimum dominating set in a series-parallel graph, which includes outerplanar graphs.  ...

### Biclique-Helly Graphs

Marina Groshaus, Jayme L. Szwarcfiter
2007 Graphs and Combinatorics
In addition, we relate biclique-Helly graphs to the classes of clique-Helly, disk-Helly and neighborhood-Helly graphs.  ...  A graph is biclique-Helly when its family of (maximal) bicliques is a Helly family. We describe characterizations for biclique-Helly graphs, leading to polynomial time recognition algorithms.  ...  In particular, he suggested the present version of the proof of Lemma 1, which is shorter and more elegant than the authors' previous proof.  ...

### Graph Isomorphism for unit square graphs [article]

Daniel Neuen
2016 arXiv   pre-print
In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time.  ...  Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.  ...  Acknowledgements I want to thank Martin Grohe and Pascal Schweitzer for several helpful discussions and comments throughout this work.  ...

### The complexity of secure domination problem in graphs

YunPing Deng, HaiChao Wang, Yancai Zhao
2018 Discussiones Mathematicae Graph Theory
In this paper, we first show that the decision version of the secure domination problem is NP-complete for star convex bipartite graphs and doubly chordal graphs.  ...  A dominating set of a graph G is a subset D ⊆ V (G) such that every vertex not in D is adjacent to at least one vertex in D.  ...  Acknowledgments The authors are grateful to the referees for their valuable comments, which result in the present version of the paper. This work was supported by the Natural  ...

### Relative neighborhood graphs and their relatives

J.W. Jaromczyk, G.T. Toussaint
1992 Proceedings of the IEEE
This is a survey of results on neighborhood graphs. The paper discusses properties, bounds on the size, algorithms, and variants of the neighborhood graphs.  ...  Numerous applications including computational morphology, spatial analysis, pattern classification, and data bases for computer vision are described.  ...  Senn for her professional help in locating many of the references.  ...

### Efficient sub-5 approximations for minimum dominating sets in unit disk graphs

Guilherme D. da Fonseca, Celina M.H. de Figueiredo, Vinícius G. Pereira de Sá, Raphael C.S. Machado
2014 Theoretical Computer Science
A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks.  ...  Because the minimum dominating set problem for unit disk graphs is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTAS.  ...  Conclusion and open problems We introduced novel efficient algorithms for approximating the minimum dominating set and minimum independent dominating set in unit disk graphs.  ...

### An Overview of Topology Mapping Algorithms and Techniques in High-Performance Computing [chapter]

Torsten Hoefler, Emmanuel Jeannot, Guillaume Mercier
2014 High-Performance Computing on Complex Environments
The chapter concludes by listing a series of open issues and problems. Introduction High Performance Computing (HPC) applications are increasingly demanding in terms of computing power.  ...  Currently, this computing power can be delivered by High-Performance Computing on Complex Environments, First edition. By Edited by Emmanuel Jeannot and JuliusŽilinskas.  ...  Acknowledgments This work is supported by the COST Action IC0805 "Open European Network for High Performance Computing on Complex Environments". Bibliography  ...

### Graph classes between parity and distance-hereditary graphs

Serafino Cicerone, Gabriele Di Stefano
1999 Discrete Applied Mathematics
Several graph problems (e.g., steiner tree, connected domination, hamiltonian path, and isomorphism problem), which can be solved in polynomial time for distance-hereditary graphs, are NP-complete or open  ...  To this purpose we characterize these classes using Cunningham decomposition and then use the devised structural characterization in order to show efficient algorithms for the recognition and isomorphism  ...  By N T (S) we denote the neighborhood of S in T , that is, the set of vertices in T that are adjacent to some vertex in S, and by N T [S] = N T (S) ∪ S the closed neighborhood of S in T .  ...

### Performance introspection of graph databases

Peter Macko, Daniel Margo, Margo Seltzer
2013 Proceedings of the 6th International Systems and Storage Conference on - SYSTOR '13
The explosion of graph data in social and biological networks, recommendation systems, provenance databases, etc. makes graph storage and processing of paramount importance.  ...  The suite has three layers of benchmarks: primitive operations, composite access patterns, and graph algorithms.  ...  We would like to thank Sparsity Technologies for providing us with a DEX license, and Dàmaris Coll Jimenez and Sergio Gómez Villamor in particular for helping us configure the database and interpret our  ...
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