How to calculate the tree depth of a regular graph, or its same order approximation when the number of vertices $n$ is large?

Question

There is a recursive definition of tree-depth in wikis:

$$ \operatorname{td}(G)=\begin{cases}1,&\text{if}|G|=1; \\\\ 1+\min\limits_{v\in V}\operatorname{td}(G-v),&\text{if } G\text{ is connected and }|G|>1; \\\\ \max\limits_i\operatorname{td}(G_i),&\text{otherwise};\end{cases} $$ Where $V$ is the set of vertices of $G$ and $G_i$ are the connected components of $G$.

Here is one simple Mathematica demonstration of the formula:

Clear["Global`*"];

(*Define the tree-depth function td according to the provided formula*)


TreeDepthOfOneCertainGraph[G_Graph] := 
  Module[{vertices, 
    connectedComponents},(*If the graph has only one vertex*)
   If[VertexCount[G] == 1, Return[1]];
   (*If the graph is connected and has more than one vertex*)
   If[ConnectedGraphQ[G],(*Compute tree-
    depth by removing each vertex and taking the minimum of the \
resulting tree-depths*)vertices = VertexList[G];
    Return[
     1 + Min[Table[
        TreeDepthOfOneCertainGraph[
         Subgraph[G, DeleteCases[vertices, v]]], {v, vertices}]]];];
   (*If the graph is not connected*)
   connectedComponents = ConnectedComponents[G];
   (*Compute the maximum tree-depth of the connected components*)
   Return[Max[
     TreeDepthOfOneCertainGraph /@ (Subgraph[G, #] & /@ 
        connectedComponents)]];];



CycleGraphsVerticesNumberList = Range[3, 10, 1];
TreeDepthOfOneCertainGraph@*CycleGraph /@ CycleGraphsVerticesNumberList
Table[2 + Ceiling[Log[2, Ceiling[(n - 2)/2] + 1]], {n, CycleGraphsVerticesNumberList}]

{3, 3, 4, 4, 4, 4, 5, 5}

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Solving the tree depth of a general graph is NP-hard, but I want to know if there are any special graphs, such as regular graphs, that can be effectively manually calculated or analyzed. For example, the tree depth of a ring with $n$ vertices (i.e. each vertex has exactly two neighbors) is $2+\lceil \log_2(\lceil \frac{n-2}{2} \rceil +1)\rceil$.

I would like to know if there are similar results (or at least estimates of order(asymptotic analysis), such as $\log_{2} n$) for a more general regular graph. Thank you!