Distinguishing number and distinguishing index of some operations on graphs

• Published in: Journal of information & optimization sciences Vol. 39; no. 5; pp. 1047 - 1059
• Main Authors: Alikhani, Saeid, Soltani, Samaneh
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 04.07.2018
• Subjects: Distinguishing index; Distinguishing number; Edge contraction
• ISSN: 0252-2667; 2169-0103
• DOI: 10.1080/02522667.2017.1294304

The distinguishing number (index) D(G) (Dʹ(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. We examine the effects on D(G) and Dʹ(G) when G is modified by operations on vertex and edge of G. Let G be a connected graph of order n ≥ 3. We show that -1 ≤ D(G - v) - D(G) ≤ D(G), where G - v denotes the graph obtained from G by removal of a vertex v and all edges incident to v and these inequalities are true for the distinguishing index. Also we prove that |D(G - e) - D(G)| ≤ 2 and -1 ≤ Dʹ (G - e) - Dʹ(G) ≤ 2, where G - e denotes the graph obtained from G by simply removing the edge e. Finally we consider the vertex contraction and the edge contraction of G and prove that the edge contraction decrease the distinguishing number (index) of G by at most one and increase by at most 3D(G) (3Dʹ(G)).