Determining number and cost of generalized Mycielskian graphs

نویسندگان

چکیده

A set $S$ of vertices is a determining for graph $G$ if every automorphism uniquely determined by its action on $S$. The size smallest called number, $Det(G)$. said to be $d$-distinguishable there coloring the with $d$ colors so that only trivial preserves color classes. such distinguishing $Dist(G)$. If $Dist(G) = 2$, cost 2-distinguishing, $\rho(G)$, class over all 2-distinguishing colorings $G$. Mycielskian, $\mu(G)$, constructed adding shadow master vertex $w$, and each $v_i$ $u_i$ edges neighborhood in $\mu(G)$ same as addition $w$. That is, $N(u_i)=N_G(v_i)\cup\{w\}$. generalized Mycielskian $\mu^{(t)}(G)$ $t$ layers vertices, above below, $w$ adjacent top layer vertices. twin-free it has no pair neighbors. This paper examines number and, when relevant, Mycielskians simple graphs isolated In particular, $G \neq K_2$ then $Det(\mu^{(t)}(G)) Det(G)$. Further, $Det(G) k \geq 2$ $t \ge k-1$, $Dist(\mu^{(t)}(G))=2$, \rho(\mu^{(t)}(G))= k$. For twins, we develop framework using quotient respect equivalence classes twin give bounds Mycielskians. Moreover, identify twins which (t{+}1)

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ژورنال

عنوان ژورنال: Discussiones Mathematicae Graph Theory

سال: 2021

ISSN: ['1234-3099', '2083-5892']

DOI: https://doi.org/10.7151/dmgt.2438