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تسجيل مستخدم جديدOn Fixing number of Functigraphs
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The fixing number of a graph $G$ is the order of the smallest subset $S$ of its vertex set $V(G)$ such that stabilizer of $S$ in $G$, $Gamma_{S}(G)$ is trivial. Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})cup V(G_{2})$ and the edge set $E(G_{1})cup E(G_{2})cup {uv:v=g(u)}$. In this paper, we study the behavior of the fixing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also study the fixing number of functigraphs of some well known families of graphs like complete graphs, trees and join graphs.
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Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})cup V(G_{2})$ and the edge set $E(G_{1})cup E(G_{2})cup {uv:g(u)=v}$. In this
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The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $Fsubseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $Pi$ $=$ ${F_1,F_2,ldots,F_k}$ be an ordered $k$-partition of $V(G)$. The
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