A $c$-labeling $\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$ of graph $G$ is distinguishing if, for every non-trivial automorphism $\pi$ of $G$, there is some vertex $v$ so that $\phi(v) \neq \phi(\pi(v))$. The distinguishing number of $G$, $D(G)$, is the smallest $c$ such that $G$ has a distinguishing $c$-labeling. We consider a compact version of Tyshkevich's graph decomposition theorem where trivial components are maximally combined to form a complete graph or a graph of isolated vertices. Suppose the compact canonical decomposition of $G$ is $G_{k} \circ G_{k-1} \circ \cdots \circ G_1 \circ G_0$. We prove that $\phi$ is a distinguishing labeling of $G$ if and only if $\phi$ is a distinguishing labeling of $G_i$ when restricted to $V(G_i)$ for $i = 0, \hdots, k$. Thus, $D(G) = \max \{D(G_i), i = 0, \hdots, k \}$. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.

翻译：$c$-标记$\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$称为图$G$的区分标记，若对$G$的每一个非平凡自同构$\pi$，存在某个顶点$v$使得$\phi(v) \neq \phi(\pi(v))$。图$G$的区分数$D(G)$是使得$G$存在区分$c$-标记的最小整数$c$。我们考虑Tyshkevich图分解定理的紧凑版本，其中平凡分量被最大程度地合并，形成完全图或孤立顶点图。假设$G$的紧凑规范分解为$G_{k} \circ G_{k-1} \circ \cdots \circ G_1 \circ G_0$。我们证明$\phi$是$G$的区分标记当且仅当$\phi$在限制到$V(G_i)$上是$G_i$的区分标记（$i = 0, \hdots, k$）。因此，$D(G) = \max \{D(G_i), i = 0, \hdots, k \}$。随后，我们提出一个在线性时间内计算单图区分数的算法。