Tree 3-Spanners on Generalized Prisms of Graphs

Gómez, Renzo; Miyazawa, Flávio Keidi; Wakabayashi, Yoshiko

doi:10.1007/978-3-031-20624-5_34

Cited by 4 publications

(2 citation statements)

References 29 publications

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“…For

GG

graphs, whose definition is very similar to

G\overline{G}

's,

t

‐admissibility is NP‐complete, for

t\ge 4

9 . Gomez et al 11 investigated spanning trees restrict to prisms of graphs, and characterized those that admit a tree 3‐spanner. As a result, they obtained a linear‐time algorithm for 3‐admissible prism graphs.…”

Section: Introductionmentioning

confidence: 99%

New parallelism and heuristic approaches for generating tree t‐spanners in graphs

Cunha,

Marciano,

Moraes

et al. 2024

Concurrency and Computation

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SummaryThe ‐admissibility is a min‐max problem that concerns to determine whether a graph contains a spanning tree in which the distance between any two adjacent vertices of is at most in . The stretch index of , , is the smallest for which is ‐admissible. This problem is in P for , NP‐complete for , , and remaining open for . In a very recent development, Couto et al. (Inf Process Lett, 2022; 177: 106265) introduced both sequential and parallel algorithms for constructing spanning trees. Additionally, they proposed two greedy heuristics for generating a candidate solution tree, but they left unresolved the issue of how to decide between two vertices when both have equal chances of being chosen in a greedy step. This criterion is important, since different branches can yield different stretch indexes. In response to this question, we developed nine new heuristics that use the concept of vertex importance in complex networks. Our research evaluates results on several types of graphs, including Barabási‐Albert, Erdős‐Rényi, Watts‐Strogatz, and bipartite graphs. Furthermore, we introduce a new parallel algorithm that employs a method using induced cycle of the graph to compare its performance with previously proposed algorithms. We develop a deep analysis on the proposed strategies (parallel and heuristics) comparing all of them to the other ones in the literature and as a result we obtain the best results so far in order to obtain exact values (or heuristics) of stretch indexes.

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“…For

GG

graphs, whose definition is very similar to

G\overline{G}

's,

t

‐admissibility is NP‐complete, for

t\ge 4

Section: Introductionmentioning

confidence: 99%

New parallelism and heuristic approaches for generating tree t‐spanners in graphs

Cunha,

Marciano,

Moraes

et al. 2024

Concurrency and Computation

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“…In 2022, Amutha A. and Mathu Pritha R., further studied the interconnection networks for the admittance of fuzziness using RL (4) . Renzo Gomez, Flavio K. Miyazawa and Yoshiko Wakabayshi derived an algorithm for the existence of tree t-spanner in an arbitrary graph (5) . Strategies for generating tree spanners were discussed by Fernanda Couto et al (6) .…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Tree t -Spanners of Ratio Labelled Butterfly Network

Pritha,

Amutha

2024

IJST

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Background/Objectives: The reliability and optimality of a communication system depend on the network that has a minimum distance. The minimum distance is the shortest path between the vertices. Even in error correction codes, the minimum Hamming distance ensures data integrity. Hence to have efficient routing protocols, the network is preferred to be a spanning tree that connects all the nodes of the communication system. As tree t- spanner is the optimization technique in determining minimum spanning tree, finding tree t-spanners of a ratio labelled fuzzy graph for the Butterfly network BF(n) is notion of this study. Methods: This study has introduced a new labelling called Ratio Labelling (RL) for examining the fuzziness in a crisp graph. Using the RL method, the crisp graphs were examined for the admittance of fuzziness and were thereby found, to have a tree t-spanners of ratio labelled fuzzy graphs. Findings: The quality of a network can be measured using the stretch factor of the network. The issues of network survival or link failures in a communication networks can be rectified by approximating the best flow spanner. As the optimal connectivity between the nodes is ensured by interconnection networks, the fuzzy tree t- spanners of undirected fuzzy Butterfly Network BF(n) are examined by labelling the vertices and edges using ratio labelling. The Breadth First Search (BFS) algorithm enhances the process of minimizing the spanning tree for an unweighted graph. Since the edge weightage is unique for all the edges of BF(n) under RL, searching for an edge with minimum weight by Kruskal’s algorithm or Prim’s algorithm are less meaningful. Hence, the BFS algorithm is adopted in finding the spanning tree T of Ratio Labelled BF(n). The bounds for the stretch factor t of a fuzzy tree t- spanner of ratio labelled BF(n) were obtained. Novelty: The literature survey shows that various classes of graphs were examined for the admissibility of tree t-spanners. Finding a tree t-spanner for a fuzzy graph was not found in literature, as per our knowledge. Extending the problem of finding a tree t -spanner of a ratio labelled fuzzy graph is an advancement in the field of fuzzy graphs. Keywords: Fuzzy Graph, Fuzzy distance, Ratio Labelling, Tree t- spanner, Butterfly Network

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