An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than $\alpha$ pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every $m$-edge graph in $Ext_{\alpha}$ can be computed in deterministic ${\cal O}(\alpha^3 m^{3/2})$ time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive $+1$-approximation of all vertex eccentricities in deterministic ${\cal O}(\alpha^2 m)$ time. This is in sharp contrast with general $m$-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in ${\cal O}(m^{2-\epsilon})$ time for any $\epsilon > 0$. As important special cases of our main result, we derive an ${\cal O}(m^{3/2})$-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an ${\cal O}(k^3m^{3/2})$-time algorithm for this problem on graphs of asteroidal number at most $k$. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.

翻译：极点是这样一个顶点：删除其闭邻域不会增加连通分量的数量。令$Ext_{\alpha}$为所有连通图的类，这些图的模分解得到的商图包含至多$\alpha$个两两不相邻的极点。我们的主要贡献如下。首先，我们证明，对于$Ext_{\alpha}$中每个有$m$条边的图，其直径可以在确定性${\cal O}(\alpha^3 m^{3/2})$时间内计算。然后，我们将运行时间改进为所有有界团数图的线性时间。此外，我们可以在确定性${\cal O}(\alpha^2 m)$时间内计算所有顶点离心率的加性$+1$近似。这与一般$m$边图形成鲜明对比：在强指数时间假设（SETH）下，对于任何$\epsilon > 0$，无法在${\cal O}(m^{2-\epsilon})$时间内计算直径。作为我们主要结果的重要特例，我们推导出一个${\cal O}(m^{3/2})$时间算法，用于计算直径至少为6的支配对图的精确直径，以及一个${\cal O}(k^3m^{3/2})$时间算法，用于星形数至多为$k$的图上的这个问题。最后，我们提出了一个改进的算法，用于有界星形数的弦图，并将我们的结果部分扩展到所有具有有界基数支配目标图的更大类。在合理的复杂性假设下，我们证明了论文中的时间上界本质上是最优的。