The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with $D(G) = d$ is called the cost of $d$-distinguishing $G$ and is denoted by $\rho_d(G)$. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of $G$, ${\rm Det}(G)$, is the minimum cardinality of determining sets of $G$. In this paper we compute the cost and the determining number for the friendship graphs and corona product of two graphs.
Alikhani, S. and Soltani, S. (2021). THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH. Journal of Algebraic Systems, 8(2), 209-217. doi: 10.22044/jas.2020.8343.1408
MLA
Alikhani, S. , and Soltani, S. . "THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH", Journal of Algebraic Systems, 8, 2, 2021, 209-217. doi: 10.22044/jas.2020.8343.1408
HARVARD
Alikhani, S., Soltani, S. (2021). 'THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH', Journal of Algebraic Systems, 8(2), pp. 209-217. doi: 10.22044/jas.2020.8343.1408
CHICAGO
S. Alikhani and S. Soltani, "THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH," Journal of Algebraic Systems, 8 2 (2021): 209-217, doi: 10.22044/jas.2020.8343.1408
VANCOUVER
Alikhani, S., Soltani, S. THE COST NUMBER AND THE DETERMINING NUMBER OF A GRAPH. Journal of Algebraic Systems, 2021; 8(2): 209-217. doi: 10.22044/jas.2020.8343.1408