# Terminology for a graph obtained from another by adding edges

$H$ is a supergraph of $G$ if it can be obtained from $G$ by adding, edges, vertices, or both.

Is there a standard terminology for a graph that can be obtained from $G$ only by adding edges?

• Sometimes it's called a closure (e.g. the transitive closure is a special case of a supergraph $H$ obtained only adding edges). However the term is only applied when you are adding all edges in order to obtain a property of the graph, so I don't think it is used as a generic term. – Bakuriu Jul 4 '16 at 11:09

If $H$ is made by adding only edges to $G$, then $G$ is a spanning subgraph of $H$. However, I'm not aware of any way of phrasing it that makes $H$ the subject of the sentence.

• "spanned supergraph" ​ ? ​ ​ ​ ​ – user12859 Jul 29 '16 at 17:02

In graph modification algorithms, a graph $H$ is called a completion of $G$ if there exists a set of edges $F \subseteq [V(G)]^2$ such that $G+F \cong H$.

You can search on arxiv for examples such as

We often think of a graph as the set of its edges, the vertex set being fixed. This is not a universal point of view, but is common enough. In that case you can just write $H \supseteq G$, and say that $H$ is a supergraph of $G$. Just make sure that you explain what you mean by these notations.