Abstract: For $2 \leq r \in \mbb{N}$, let $S_r$ denote the class of graphs consisting of subdivisions of the wheel graph with $r$ spokes in which the spoke edges are left undivided. Interest is in structure and size of graphs containing no $S_r$-subgraph. Let $ex(n,S_r)$ denote the maximum number of edges of a graph containing no $S_r$-subgraph, and let $Ex(n,S_r)$ denote the class of all $n$-vertex graphs containing no $S_r$-subgraph that are of size $ex(n,S_r)$. A conjecture is put forth stating that for $r \geq 3$ and $n \geq 2r+1$, $ex(n,S_r) = (r-1) n - \left\lceil (r-1)(r-3/2) \right\rceil$ and for $r \geq 4$ $Ex(n,S_r)$ consists of a single graph which is the graph obtained from $K_{r-1,n-r+1}$ by adding a maximum matching to the color class of cardinality $r-1$. A previous result of C. Thomassen asserts this conjecture holds for $r=3$. We show that it holds for $r = 4$. More generally, for $r \geq 4$, it is shown that $ex(n,S_r) = \Theta(rn)$ and thus essentially resolving a problem of Bollob{\' a}s. If time permits, we shall discuss structure of $r$-connected graphs containing no $S_r$-subgraph. Some of the results presented in this talk are a joint work with M. Lomonosov.