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To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jetson $X$. Its fiber over a singular point of $X$ is called a singular fiber. Fora surface with a rational double point, Mourtada gave a one-to-onecorrespondence between the irreducible components of the singular fiber of$X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg0$.  In this paper, for a surface $X$ over complex number with a singularity of$A_n$ or $D_4$-type, we study the intersections of irreducible components ofthe singular fiber and construct a graph using this information. The verticesof the graph correspond to irreducible components of the singular fiber and twovertices are connected when the intersection of the corresponding components ismaximal for the inclusion relation. In the case of $A_n$ or $D_4$-typesingularity, we show that this graph is isomorphic to the resolution graph for$m \gg 0$.

On the configuration of the singular fibers of jet schemes of rational double points