# Symbol for Graph Difference?

Is there any well-defined symbol to denote the difference between the two graphs. The difference between two graphs $$G$$ and $$H$$ is defined as the remaining sub-graph $$G'$$ of $$G$$ after the subgraph $$H$$ is removed from $$G$$ (assuming $$H$$ is a sub-graph of $$G$$). For example (the image is taken from Wolfram):

Note that, $$G'$$ might not be unique, since $$H$$ can be positioned anywhere in $$G$$, Although I can define my own symbol, it would be better to use the symbol that is well defined by the community.

• given that union and complement are well defined, I think the natural suggestion is to use the \setminus for difference en.wikipedia.org/wiki/Graph_operations Aug 16 '20 at 5:23
• @DanielS. \setminus is for set difference right? So we can not use it for a graph $G = (V,E)$.
– IY3
Aug 16 '20 at 6:30

If you use notation $$G-H$$ or $$G\setminus H$$, it will often be interpreted as taking the graph $$G$$ and removing from it all the vertices of $$H$$ and any edge incident with a vertex of $$H$$. Which is not what you want.

The easiest way to write this would probably be $$G-E(H)$$, as this makes it clear you are only removing edges, and is unambiguous.

Alternatively, you could just define your own symbol / notation and state what it means up front, this is perfectly acceptable so long as you give a clear definition.

Based on the answer of @Brandon du Preez, we can also define the graph difference between $$G$$ and $$H$$, as $$G'= (V',E')$$, such that:

$$V' = V(G)$$, and $$E' = E(G)-E(H)$$