How to graph a monopoly that does not exhibit the properties of a contestable market?
The diagram above describes a Natural Monopoly. These economies have the characteristic that the long run average costs are decreasing (and hence the LRMC is below the LRAC curve). That is, costs reduce as the firm keeps producing more. *Firms with network effects (e.g. telecom industry) is an example of such an industry.

Lets look at 2 cases: Competitive market and a single firm.

If the market is competitive, the firms will produce at a price-quantity pair such that $LRMC =P$. But at this price, the total revenue is negative (since the firm earns 0 profit at $(Q_C,P_C)$).
Thus the firms will be better off shutting down and earning 0 profits. Clearly, a competitive market won't work in such cases.

For a monopoly, the firm will charge price $P_M$ and produce $Q_M$ (the point where $LRMC=MR$). This firm earns a positive profit ($Q_MP_M&gt;LRAC*Q_M$).