# On Resolving Efficient Dominating Set of Cycle and Comb Product Graph

Authors
Muzayyanatun Munawwarah1, Dafik1, 2, *, Arika Indah Kristiana1, 2, Elsa Yuli Kurniawati2, Rosanita Nisviasari2
1Departement of Postgraduate Mathematics Education, University of Jember, Jember, Indonesia
2PUI-PT Combinatorics and Graph, CGANT, University of Jember, Jember, Indonesia
*Corresponding author. Email: d.dafik@unej.ac.id
Corresponding Author
Dafik
Available Online 27 April 2023.
DOI
10.2991/978-94-6463-138-8_2How to use a DOI?
Keywords
Resolving Efficient Dominating Set; Cycle Graph; Comb Product Graph
Abstract

The graph used in this paper is a connected, bounded, and undirected graph G, is used which contains a set of vertex V(G) and a set of edge E(G). It is called the efficient dominating set of a graph if every point V in D or is adjacent to one vertex in D. For a set of solutions of G in an ordered set, it is distinguished by the distance of its point representation. Suppose we take any vertex in G, then S = s 1 , s 2 , . . . , s k is a subset of V(G) and the ordered set W of vertex representation is r ( p | S ) = ( d ( p , s 1 ) , d ( p , s 2 ) , . . . , d ( p , s k ) ) . The set S can be is called the completion set of G if r ( u | S ) r ( p | S ) u , v G . And for subset Z of V(G) it can be called the efficient dominating set if r ( u | Z ) r ( p | Z ) u , v G , then the minimum cardinality of resolving efficient dominating set is symbolized by γ re ( G ) . The axiomatic deductive technique and the pattern detection method used in this study apply the principles of deductive proof to mathematical logic by using existing axioms, lemmas, and theorems to solve questions about the topic under study. Some theorems or definitions will be obtained in this study as a result of further analysis of previously existing theorems or definitions. The pattern identification approach follows a research method for locating efficient set completion patterns in the graph under consideration and the problem. In this paper we obtain γ re ( G ) from several cycle graphs G m C n , namely P m C n , F m C n , K l , m C n , in this paper the proving of resolving efficient dominating set is only on n 0 ( m o d 3 ) .

Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

Volume Title
Proceedings of the 6th International Conference of Combinatorics, Graph Theory, and Network Topology (ICCGANT 2022)
Series
Advances in Physics Research
Publication Date
27 April 2023
ISBN
10.2991/978-94-6463-138-8_2
ISSN
2352-541X
DOI
10.2991/978-94-6463-138-8_2How to use a DOI?
Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

### Cite this article

TY  - CONF
AU  - Muzayyanatun Munawwarah
AU  - Dafik
AU  - Arika Indah Kristiana
AU  - Elsa Yuli Kurniawati
AU  - Rosanita Nisviasari
PY  - 2023
DA  - 2023/04/27
TI  - On Resolving Efficient Dominating Set of Cycle and Comb Product Graph
BT  - Proceedings of the 6th International Conference of Combinatorics, Graph Theory, and Network Topology (ICCGANT 2022)
PB  - Atlantis Press
SP  - 3
EP  - 16
SN  - 2352-541X
UR  - https://doi.org/10.2991/978-94-6463-138-8_2
DO  - 10.2991/978-94-6463-138-8_2
ID  - Munawwarah2023
ER  -