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Search Results (5)

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Keywords = clique graph of a chordal graph

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26 pages, 1131 KiB  
Article
Perfect Roman Domination: Aspects of Enumeration and Parameterization
by Kevin Mann and Henning Fernau
Algorithms 2024, 17(12), 576; https://doi.org/10.3390/a17120576 - 14 Dec 2024
Cited by 1 | Viewed by 1035
Abstract
Perfect Roman Dominating Functions and Unique Response Roman Dominating Functions are two ways to translate perfect code into the framework of Roman Dominating Functions. We also consider the enumeration of minimal Perfect Roman Dominating Functions and show a tight relation to minimal Roman [...] Read more.
Perfect Roman Dominating Functions and Unique Response Roman Dominating Functions are two ways to translate perfect code into the framework of Roman Dominating Functions. We also consider the enumeration of minimal Perfect Roman Dominating Functions and show a tight relation to minimal Roman Dominating Functions. Furthermore, we consider the complexity of the underlying decision problems Perfect Roman Domination and Unique Response Roman Domination on special graph classes. For instance, split graphs are the first graph class for which Unique Response Roman Domination is polynomial-time solvable, while Perfect Roman Domination is NP-complete. Beyond this, we give polynomial-time algorithms for Perfect Roman Domination on interval graphs and for both decision problems on cobipartite graphs. However, both problems are NP-complete on chordal bipartite graphs. We show that both problems are W[1]-complete if parameterized by solution size and FPT if parameterized by the dual parameter or by clique width. Full article
(This article belongs to the Special Issue Selected Algorithmic Papers from IWOCA 2024)
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20 pages, 593 KiB  
Article
Properties and Recognition of Atom Graphs
by Geneviève Simonet and Anne Berry
Algorithms 2022, 15(8), 294; https://doi.org/10.3390/a15080294 - 19 Aug 2022
Viewed by 2184
Abstract
The atom graph of a connected graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all its atom trees. A graph G is an atom graph if [...] Read more.
The atom graph of a connected graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all its atom trees. A graph G is an atom graph if there is a graph whose atom graph is isomorphic to G. We study the class of atom graphs, which is also the class of atom graphs of chordal graphs, and the associated recognition problem. We prove that each atom graph is a perfect graph and give a characterization of atom graphs in terms of a spanning tree, inspired by the characterization of clique graphs of chordal graphs as expanded trees. We also characterize the chordal graphs having the same atom and clique graph, and solve the recognition problem of atom graphs of two graph classes. Full article
(This article belongs to the Special Issue Combinatorial Designs: Theory and Applications)
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18 pages, 440 KiB  
Article
Remarks on Parameterized Complexity of Variations of the Maximum-Clique Transversal Problem on Graphs
by Chuan-Min Lee
Symmetry 2022, 14(4), 676; https://doi.org/10.3390/sym14040676 - 24 Mar 2022
Cited by 5 | Viewed by 2019
Abstract
With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. Backhaul requirements within current wireless networks are [...] Read more.
With the rapid growth in the penetration rate of mobile devices and the surge in demand for mobile data services, small cells and mobile backhaul networks have become the critical focus of next-generation mobile network development. Backhaul requirements within current wireless networks are almost asymmetrical, with most traffic flowing from the core to the handset, but 5G networks will require more symmetrical backhaul capability. The deployment of small cells and the placement of transceivers for cellular phones are crucial in trading off the symmetric backhaul capability and cost-effectiveness. The deployment of small cells is related to the placement of transceivers for cellular phones. Chang, Kloks, and Lee transformed the placement problem into the maximum-clique transversal problem on graphs. From the theoretical point of view, our paper considers the parameterized complexity of variations of the maximum-clique transversal problem for split graphs, doubly chordal graphs, planar graphs, and graphs of bounded treewidth. Full article
(This article belongs to the Special Issue Graph Algorithms and Graph Theory)
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14 pages, 497 KiB  
Article
Algorithmic Aspects of Some Variations of Clique Transversal and Clique Independent Sets on Graphs
by Chuan-Min Lee
Algorithms 2021, 14(1), 22; https://doi.org/10.3390/a14010022 - 13 Jan 2021
Cited by 7 | Viewed by 2931
Abstract
This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly [...] Read more.
This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal graphs, clique perfect graphs, total graphs, split graphs, line graphs, and dually chordal graphs. We give equations to compute the {k}-clique, minus clique, signed clique, and k-fold clique transversal numbers for suns, and show that the {k}-clique transversal problem is polynomial-time solvable for graphs whose clique transversal numbers equal their clique independence numbers. We also show the relationship between the signed and generalization clique problems and present NP-completeness results for the considered problems on k-trees with unbounded k, planar graphs, doubly chordal graphs, total graphs, split graphs, line graphs, and dually chordal graphs. Full article
(This article belongs to the Special Issue Graph Algorithms and Applications)
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23 pages, 311 KiB  
Article
Computing a Clique Tree with the Algorithm Maximal Label Search
by Anne Berry and Geneviève Simonet
Algorithms 2017, 10(1), 20; https://doi.org/10.3390/a10010020 - 25 Jan 2017
Cited by 4 | Viewed by 5276
Abstract
The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) [...] Read more.
The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph. Full article
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