Tyshkevich图分解与单图的区分数
Tyshkevich's Graph Decomposition and the Distinguishing Numbers of Unigraphs
AI总结：
该研究利用Tyshkevich图分解的紧凑形式，证明了图的区分数等于其各规范分解分量区分数的最大值，并提出了线性时间计算单图区分数的算法。
摘要：
图$G$的一个$c$-标记$\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$是可区分的，当且仅当对$G$的每个非平凡自同构$\pi$，都存在某个顶点$v$使得$\phi(v) \neq \phi(\pi(v))$。$G$的区分数$D(G)$是满足$G$存在可区分$c$-标记的最小$c$。\n 我们研究了Tyshkevich图分解定理的一个紧凑版本，其中平凡分量被极大地组合以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$G$的可区分标记当且仅当对于$i = 0, \hdots, k$，$\phi$限制在$V(G_i)$上时是$G_i$的可区分标记。因此，$D(G) = \max \{D(G_i), i = 0, \hdots, k \}$。随后我们提出了一种在线性时间内计算单图（unigraph）区分数的算法。
英文摘要：
A $c$-labeling $ϕ: V(G) \rightarrow \{1, 2, \hdots, c \}$ of graph $G$ is distinguishing if, for every non-trivial automorphism $π$ of $G$, there is some vertex $v$ so that $ϕ(v) \neq ϕ(π(v))$. 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 $ϕ$ is a distinguishing labeling of $G$ if and only if $ϕ$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
