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# Nearly Complete Graph

A graph is a collection of vertices and edges. A graph is complete if there is an edge connecting every vertex to every other vertex. A graph is nearly complete if it can be obtained by removing a small number of edges from a complete graph relative to the size of the graph.

1. Mathematical Definition Consider a graph with vertices v, edges e, and genus g. Calculate

```bash
: X = (e - 3v + 6)/6 .
```

Let \lceil X \rceil denote the smallest integer greater than or equal to X. All graphs satisfy Euler's lower bound : g \geq \lceil X \rceil . For complete graphs g =\lceil X \rceil and the bound is saturated. One may start with a complete graph and remove p edges such that the remaining graph satisfies

* Euler's lower bound is saturated g =\lceil X \rceil
* The graph is connected Let NC(n) denote the maximum number of possible edge removals from the complete graph K\_n such that the above two properties hold no matter which edges are removed. A graph with n vertices is **nearly complete** if it can be obtained by removing p \leq NC(n) edges from the complete graph K\_n.

1. References

* [The genus of nearly complete graphs-case 6, Jonathan L. Gross, Aeq. Math. 13, 243–249 (1975) doi:10.1007/BF01836527](https://link.springer.com/article/10.1007/BF01836527)


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