TY - JOUR
T1 - Local Multiset Dimension of Amalgamation Graphs
AU - Alfarisi, Ridho
AU - Susilowati, Liliek
AU - Dafik, Dafik
AU - Prabhu, Savari
N1 - Funding Information:
This work was supported in part by University of Jember and Universitas Airlangga, Indonesia.
Funding Information:
We gratefully acknowledge the support from Universitas Airlangga and University of Jember of year 2022.
Publisher Copyright:
Copyright: © 2023 Alfarisi R et al.
PY - 2023
Y1 - 2023
N2 - \textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension.
AB - \textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension.
KW - local m-resolving set; local multiset dimension; amalgamation graph
UR - http://www.scopus.com/inward/record.url?scp=85152672591&partnerID=8YFLogxK
U2 - 10.12688/f1000research.128866.1
DO - 10.12688/f1000research.128866.1
M3 - Article
AN - SCOPUS:85152672591
SN - 2046-1402
VL - 12
JO - F1000Research
JF - F1000Research
M1 - 95
ER -