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- W4300193232 abstract "The distinguishing number (index) $D(G)$ ($D'(G)$) 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 $D'(G)$ when $G$ is modified by operations on vertex and edge of $G$. Let $G$ be a connected graph of order $ngeq 3$. We show that $-1leq D(G-v)-D(G)leq D(G)$, 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 $|D(G-e)-D(G)|leq 2$ and $-1 leq D'(G-e)-D'(G)leq 2$, 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 $3D(G)$ ($3D'(G)$)." @default.
- W4300193232 created "2022-10-03" @default.
- W4300193232 creator A5029127053 @default.
- W4300193232 creator A5056899761 @default.
- W4300193232 date "2016-05-23" @default.
- W4300193232 modified "2023-09-27" @default.
- W4300193232 title "Distinguishing number and distinguishing index of some operations on graphs" @default.
- W4300193232 doi "https://doi.org/10.48550/arxiv.1605.07016" @default.
- W4300193232 hasPublicationYear "2016" @default.
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