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Let $f: X \to \Delta$ be a flat, projective morphism, smooth over the punctured disc $\Delta^*:=\Delta \backslash \{0\}$ and central fiber $f^{-1}(0)$ is a reduced, simple normal crossings divisor. Does there exist a Whitney stratification of $X$ with one strata consisting of $f^{-1}(\Delta^*)$? For example, in the case $X$ is of relative dimension $1$ (i.e., $X$ is a family of curves) and the central fiber is the union of two smooth, irreducible curves $Y_1, Y_2$, is $$\{f^{-1}(\Delta^*)\} \coprod \{(Y_1 \cup Y_2)\backslash (Y_1 \cap Y_2)\} \coprod \{Y_1 \cap Y_2\}$$ a Whitney stratification of $X$? Any idea/reference will be most welcome.

EDIT: Assume $X$ is a smooth, complex manifold.

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