In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter $\tau$. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for $0 \leq \tau \leq 2\varepsilon$, where $\varepsilon$ is the smoothing parameter. Then, for the restriction of $\tau \in [0,\varepsilon]$, we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope $m \in [0,1]$. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every $m \in [0,1]$, which is a generalization of the original interleaving distance, which is the case $m=0$. While the resulting metrics are not stable, we show that any pair of these for $m,m' \in [0,1)$ are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.

翻译：在本文中, 我们在 Reeb 图形上推出一个“ 平滑” 的扩展, 我们称之为平滑平滑的平滑; 这反过来让我们可以定义一个新的量子组, 将Reeb 图形的间距宽化。 直观地说, 在平滑过程中, 我们“ 切开” 部分在本地迷你和最大值之间, 削减量受一个参数 $\ tau 控制 。 在将电流正规化后, 我们显示, 在平滑的调试器应用后, 这会阻碍功能范围的广泛扩展, 并产生特别不错的特性( 例如保持连通性) 。 当与平滑的 $\ leq\ tau\ leq\ \ leq 2\ varepsilon 相加在一起时, 我们“ 平滑” 平滑的量值在平滑时, 我们可以用额外的结构来构建一个直直线流, 在每平坦值 $ 0. 0, 美元 的平流 内, 内, 以每平流 平流 平流 平流 的平流 。 平流 平流 平流 平流 平流 直径 直径 直径 显示 直径 直径 直径 直 直 直 直 直 直径 直径 直径 直径 直 直 直 直径 直径 直 直 直 直径 。