Distinguishing Number of the Generalized Theta Graph

Abstract

A generalized theta graph is a graph constructed from two distinct vertices by joining them with 𝑙 (>=3) internally disjoint paths of lengths greater than one. The distinguishing number 𝐷(𝐺) of a graph 𝐺 is the least integer 𝑑 such that 𝐺 has a vertex labelling with 𝑑 labels that is preserved only by a trivial automorphism. The partition dimension of a graph G is the least k such that V(G) can be k-partitioned such that the representations of all vertices are distinct with respect to that partition. In this paper, we establish a relation between the distinguishing number and the partition dimension of a graph. We also determine the distinguishing number for the generalized theta graph.