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：设 $G$ 为图，一个 $c$ 标签 $\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$ 被称为可区分的，如果对于 $G$ 的每个非平凡自同构 $\pi$，存在某个顶点 $v$ 使得 $\phi(v) \neq \phi(\pi(v))$。图 $G$ 的可区分数 $D(G)$ 是使得 $G$ 存在可区分 $c$ 标签的最小 $c$。我们考虑蒂什凯维奇图分解定理的一个紧凑版本，其中平凡分量被最大程度地合并，从而形成完全图或孤立顶点图。假设 $G$ 的紧凑典范分解为 $G_{k} \circ G_{k-1} \circ \cdots \circ G_1 \circ G_0$。我们证明，$\phi$ 是 $G$ 的可区分标签当且仅当对于 $i = 0, \hdots, k$，$\phi$ 限制在 $V(G_i)$ 上是 $G_i$ 的可区分标签。因此，$D(G) = \max \{D(G_i), i = 0, \hdots, k \}$。随后，我们提出一种在线性时间内计算唯一图可区分数的算法。