A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is connected. The connected domination number of $G$, $\gamma_c(G)$, is the minimum cardinality of a connected dominating set of $G$. A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv) < k$ for every pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut-vertices of $G$. It is known that if $G$ is a $k$-$\gamma_{c}$-critical graph, then $G$ has at most $k - 2$ cut-vertices, that is $\zeta \le k - 2$. In this paper, for $k \ge 4$ and $0 \le \zeta \le k - 2$, we show that every $k$-$\gamma_{c}$-critical graph with $\zeta$ cut-vertices has a hamiltonian path if and only if $k - 3 \le \zeta \le k - 2$.

中文翻译：

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连通支配临界图的可追溯性

图$G$ 中的支配集是$G$ 顶点的集合$S$，使得$S$ 之外的每个顶点都与$S$ 中的顶点相邻。$G$中的连通支配集是支配集$S$，使得$S$诱导的子图$G[S]$是连通的。$G$的连通支配数，$\gamma_c(G)$，是$G$连通支配集的最小基数。如果连通支配数 $\gamma_{c}(G)$ 等于 $k$ 和 $\gamma_{c}，则称图 $G$ 是 $k$-$\gamma_{c}$-critical (G + uv) < k$ 对于 $G$ 的每对非相邻顶点 $u$ 和 $v$。令 $\zeta$ 为 $G$ 的割点数。已知如果$G$是$k$-$\gamma_{c}$-临界图，则$G$最多有$k - 2$个割点，即$\zeta \le k - 2美元。在本文中，对于 $k \ge 4$ 和 $0 \le \zeta \le k - 2$，