The existence of planar distance preserver? Let G be an n-node undirected graph, and let T be a node subset of V(G) called terminals.  A distance preserver of (G,T) is a graph H satisfying the property 

$$d_H(u,v) = d_G(u,v)$$ 

for all nodes u, v in T. (Note that H is NOT necessarily a subgraph of G.)

For example, let G be the following graph (a) and T be the nodes on the external face. Then graph (b) is a distance preserver of (G,T).



Distance preserver with various parameters are known to exist. I'm particularly interested in the one with the following properties:


G is planar and unweighted (that is, all edges of G has weight one),
T has size $O(n^{0.5})$, and
H has size (the number of nodes and edges) $o(n)$. (It would be nice if we have $O(\frac{n}{\log\log n})$.)



  Does such a distance preserver exist?


If one cannot meet the above properties, any kind of relaxations are welcomed.



References:


Sparse Source-wise and Pair-wise Distance Preservers, Don Coppersmith and Michael Elkin, SIDMA, 2006.
Sparse Distance Preservers and Additive Spanners, Béla Bollobás, Don Coppersmith, and Michael Elkin, SIDMA, 2005.
Spanners and emulators with sublinear distance errors, Mikkel Thorup and Uri Zwick, SODA, 2006.
Lower Bounds for Additive Spanners, Emulators, and More, David P. Woodruff, FOCS, 2006.


Distance preserver is also known as an emulator; many related work can be found on internet by searching the term spanner, which requires H to be a subgraph of G. But in my applications we can use other graphs as well, as long as H preserves the distances between T in G.
Many years later, it looks like OP has finally answered his own question: Near-Optimal Distance Emulator for Planar Graphs by Hsien-Chih Chang, Paweł Gawrychowski, Shay Mozes, and Oren Weimann was just posted on the arxiv.

The answer to the original question is yes: it is shown that $\widetilde{O}(\min\{t^2, \sqrt{tn}\})$ edges suffice to preserve distances between $|T| =: t$ terminals, which is optimal up to log factors.  In particular $\widetilde{O}(n^{3/4})$ edges suffice for the setting in the OP.  This preserver can also be computed in $\widetilde{O}(n)$ time; I would strongly suspect that the log factors in the size can be removed if we care only about existence and not computation time of the preserver, but I have not rigorously verified this.

(On a less formal note, I find this result really amazing.  Congrats!)