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The Bondy-Chvatal closure of a finite simple graph
Definition
Let be a finite simple graph on vertices. Repeatedly add an edge between nonadjacent vertices whenever their degrees in the current graph satisfy
Stop when no such pair remains. The resulting graph is the Bondy-Chvatal closure of , denoted . The edge addition in If nonadjacent in an -vertex graph satisfy , then adding preserves Hamiltonicity in both directions is the operation used at each stage. The fact that the terminal graph does not depend on the order of eligible additions is The Bondy-Chvatal closure is independent of the order of eligible edge additions ↗.
Depends on
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- If nonadjacent $u,v$ in an $n$-vertex graph satisfy $\deg(u)+\deg(v)\ge n$, then adding $uv$ preserves Hamiltonicity in both directions
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 14 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Applied Combinatorics, Eulerian and Hamiltonian Graphs (standard reference, not scraped)