Proceedings of the 6th International Conference of Combinatorics, Graph Theory, and Network Topology (ICCGANT 2022)

# On the Resolving Efficient Domination Number of Path and Comb Product of Special Graph

Authors
Chalim Mufidah1, 2, Dafik1, 2, Arika Indah Kristiana1, 2, *, Slamin2, 3, Ika Hesti Agustin2, 4
1Department of Mathematics Education Postgraduate, University of Jember, Jember, Indonesia
2PUI-PT Combinatorics and Graph, CGANT, University of Jember, Jember, Indonesia
3Department of Computer Science, University of Jember, Jember, Indonesia
4Department of Mathematics, University of Jember, Jember, Indonesia
*Corresponding author.
Corresponding Author
Arika Indah Kristiana
Available Online 27 April 2023.
DOI
10.2991/978-94-6463-138-8_9How to use a DOI?
Keywords
Resolving Efficient Dominating Number; Path Graph; Comb Product of Special Graph
Abstract

Suppose G is a bounded, connected, and undirected graph. Let V(G) and E(G) be the vertex set and edge set, respectively. The subset D of V(G) is a subset of the graph G that dominates it efficiently if every vertex in G is either in D or adjacent to a vertex in D. The subset M of V(G) is the complete set of G if every vertex in G is differentiated differently based on its representation to each vertex in the ordered set M. Suppose M = { b 1 , b 2 , b 3 , , b k } is a subset of V(G). The representation of a vertex n G on an ordered set M is r ( n | M ) = { ( d ( n , b 1 ) , d ( n , b 2 ) , d ( n , b 3 ) , d ( n , b k ) } . A set M is called a set of solutions of G if r ( u | M ) r ( n | M ) u , n G . The subset Z of V(G) is called the set of efficient dominating solutions of a graph G if is an efficient dominating set and r ( u | M ) r ( n | M ) u , n G . Let γ re ( G ) denote the minimum cardinality of an efficient domination number. We call it efficient dominance number resolution on graphs. The research methods used in this research are the axiomatic deductive method and the pattern detection method. The axiomatic deductive method is a research method that applies the principles of deductive proof (from general to specific) that apply to mathematical logic by using axioms, lemmas, and existing theorems to solve a problem on the topic under study. In this research, some theorems or definitions will be obtained from the results of further analysis of some previously existing theorems or definitions. Next is the pattern detection method, which is a research method to find efficient set completion patterns in the graph under study and the problem to be solved. In this article, the efficiency solution of the γ re G domination number of several will be obtained comb product graph is B F a P b , T S a P b , and L a P b .

Open Access

Volume Title
Proceedings of the 6th International Conference of Combinatorics, Graph Theory, and Network Topology (ICCGANT 2022)
Series
Publication Date
27 April 2023
ISBN
10.2991/978-94-6463-138-8_9
ISSN
2352-541X
DOI
10.2991/978-94-6463-138-8_9How to use a DOI?
Open Access

TY  - CONF
AU  - Chalim Mufidah
AU  - Dafik
AU  - Arika Indah Kristiana
AU  - Slamin
AU  - Ika Hesti Agustin
PY  - 2023
DA  - 2023/04/27
TI  - On the Resolving Efficient Domination Number of Path and Comb Product of Special Graph
BT  - Proceedings of the 6th International Conference of Combinatorics, Graph Theory, and Network Topology (ICCGANT 2022)
PB  - Atlantis Press
SP  - 94
EP  - 106
SN  - 2352-541X
UR  - https://doi.org/10.2991/978-94-6463-138-8_9
DO  - 10.2991/978-94-6463-138-8_9
ID  - Mufidah2023
ER  -