比特币(BSV)知识库：比特币原理-近完全图(Nearly Complete Graph)

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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.

Mathematical Definition

Consider a graph with vertices , edges , and genus .

Calculate

.

Let  denote the smallest integer greater than or equal to . All graphs satisfy Euler's lower bound

  .

For complete graphs  and the bound is saturated.

One may start with a complete graph and remove  edges such that the remaining graph satisfies

• Euler's lower bound is saturated 
• The graph is connected

Let  denote the maximum number of possible edge removals from the complete graph  such that the above two properties hold no matter which edges are removed. A graph with  vertices is nearly complete if it can be obtained by removing  edges from the complete graph .

References

• The genus of nearly complete graphs-case 6, Jonathan L. Gross, Aeq. Math. 13, 243–249 (1975) doi:10.1007/BF01836527

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