Q:

Fruit carving on Shabbat? 

There is a technique called Fruit carving, consisting on cutting food into a pleasant shape. 
Is it permitted to carve such things on Shabbat? ( And on Yom Tov)
Reflection may make a difference between giving a concrete shape, or just an abstract one. 

A:

Shemiras Shabbos Kehilchoso 11 (12) says one should not use a special utensil to cut a melon in a special shape, like a triangle. ( Nor any other utensil to cut potatoes).  He quotes the Mishnah Beruro and Shaar Hatziyun: 500 MB 17; ShH (20). 

….......... אבל אותיות [וציורים] אסור לעשותן בבשר דרך סימן ביו"ט: 
But letters and pictures are forbidden to be made in meat as a sign on
  Yom Tov. 

The key parts of the long  Shaar Hatziyun are:
To make an impression like a shape is certainly forbidden on Yom Tov, and even non-permanent writing  which is a Rabbinic prohibition was not permitted on Yom Tov because of the permission to make food on Yom Tov. ( Only the melochos from “kneading” and onwards are permitted).  The same applies to the other 39 melochos in which there is no difference between Shabbos and Yom Tov.  

The Shemiras Shabbos seems to limit the prohibition to a special
utensil.  This is not fully evident to me from his sources. 

In “The 39 Melochos” by Rabbi Dovid Ribiat, Chapter 11, Kosaiv, he writes,  p 951, 
in b] that by molding or cutting out letters and shapes that Kosaiv can occur even if there is no surface or backing on which the writing occurs. 
On p 952, he gives an example of cutting vegetables into a flower or other shapes on Shabbos or cutting a melon into a basket or boat-like shape.  
He quotes the Chayyei Odom Hilchos Yom Tov 92 (3) 

אבל אסור לחקוק בדפוס בעיסה איזה צורה.  ונ"ל דאפי' ביד אסור לעשות מן
  העיסה כמין צורת עוף או שאר צורה: 
but it is forbidden to use a press to create any form from dough.  And
  it seems to me that even to create the form of a bird or other thing
  by hand from the dough will also be forbidden. 

and the Gemoro Shabbos 104a which says that the final letter Mem and the letter Samech in the Tablets were visible from both sides of the tablets and were hanging in the air, which establishes that no background is needed to create a letter.  

There is more discussion in the Hebrew note 27.  He evidently sees the
Chayyei Odom's mention of dough as being just an example. 

Conclusion.  There is certainly a potential prohibition here.  Meaningful patterns or signs seem to be more problematic.  CYLOR. 