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- W3196653874 abstract "A vertex coloring of a graph $G$ is distinguishing if any non-identity automorphism of $G$ does not preserve it. The distinguishing number is the minimum number of colors required for such a coloring and the distinguishing threshold is the minimum number of colors~$k$ such that any arbitrary $k$-coloring is distinguishing. In this paper, we consider the distinguishing number and the distinguishing threshold for some graph operations, namely the vertex-sum of graphs as well as the rooted, corona and lexicographic product of graphs. As an important tool, we define a steady vertex of a graph as a vertex $v$ for which we have $mathrm{Aut}(G-v)congmathrm{stab}_{mathrm{Aut}(G)}(v)$. Using this notion, we prove that a vertex $v$ is steady if and only if any distinguishing coloring for $G$ induces a distinguishing coloring for $G-v$." @default.
- W3196653874 created "2021-09-13" @default.
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- W3196653874 date "2021-08-31" @default.
- W3196653874 modified "2023-09-26" @default.
- W3196653874 title "Breaking symmetries of some graph operations by vertex coloring" @default.
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