TY - JOUR

T1 - On commutative characterization of graph operation with respect to metric dimension

AU - Susilowati, Liliek

AU - Utoyo, Mohammad Imam

AU - Slamin,

N1 - Publisher Copyright:
© 2017 Published by ITB Journal Publisher.

PY - 2017

Y1 - 2017

N2 - Let G be a connected graph with vertex set V(G) and W={w1, w2, …, wm} ⊆ V(G). A representation of a vertex v ∈ V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),…,d(v,wm)) where d(v,w) is the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.

AB - Let G be a connected graph with vertex set V(G) and W={w1, w2, …, wm} ⊆ V(G). A representation of a vertex v ∈ V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),…,d(v,wm)) where d(v,w) is the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.

KW - Comb product

KW - Commutative with respect to metric dimension

KW - Corona product

KW - Generalized comb and corona products

KW - Metric dimension basis

UR - http://www.scopus.com/inward/record.url?scp=85030649705&partnerID=8YFLogxK

U2 - 10.5614/j.math.fund.sci.2017.49.2.5

DO - 10.5614/j.math.fund.sci.2017.49.2.5

M3 - Article

AN - SCOPUS:85030649705

SN - 2337-5760

VL - 49

SP - 156

EP - 170

JO - Journal of Mathematical and Fundamental Sciences

JF - Journal of Mathematical and Fundamental Sciences

IS - 2

ER -