77 lines
3.3 KiB
Markdown
77 lines
3.3 KiB
Markdown
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---
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title: Graph Theory
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updated: 2022-07-18 09:46:19Z
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created: 2021-10-02 17:39:39Z
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latitude: 52.09370000
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longitude: 6.72510000
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altitude: 0.0000
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---
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## Euler Path
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uses every single egde once; no repeats
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No need to have start en end vertex the same.
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**0 or 2 odd degree vertices, the rest even**
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## Euler Circuit
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Every edge once; no repeats
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Start and end vertex are the same
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**All vertices must have an even degree**
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Multiple Euler Circuits are possible and depends on the starting direction.
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## Fleury's Algorithme
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Find an Euler Circuit, starting at vertex A
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Remove the edge take to go the next vertex, but a disconnected graph is not allowed. Deleting an edge prevents backtracking.
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## Eulerization
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Duplicate egdes so a Euler cirquit is in the graph, from and to vertices with odd degree. No new edges. Minimize duplication of edges. Multiple solutions possible.
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## Hamiltonian Circuit
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Visit every vertex once, with no repeats,
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Minimun cost Hamiltonian Circuit == (Traveling Salesman Problem)
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Possible method: Brute Force
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Optimal, bit not efficient
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## Complete Graph
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All vertices are connected with all the others.
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If n vertices then (n-1)!/2 different hamiltonian circuits
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Better then Brute Force the get the shortest circuit are heuristic methods;
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- **Nearest Neighbor Algorithm** (not optimal)
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Start at A and take at every vertex the edge with the lowest weight. (Greedy, doesn't look ahead)
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- **Repeated Nearest Algorithm**
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Same as Nearest Neighbor Algorithm but repeat for every vertex and select the lowest cost.
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- **Sorted Edge Algorithm** (not optimal)
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Add cheapest up, unless:
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- a mini circuit (a curcuit that doesn't include all vertices)
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- no vertex with a degree 3
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AD = 1
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AC = 2
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AB = 4 (not because of degree 3 rule)
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CD = 8 (not because creates mini circuit)
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BD = 9
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BC = 13
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Most optimal is ADBCA = 25
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## Hamiltonian Path
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Visit every vertex once, with no repeats,
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## Kruskal's Algoritme (optimal and efficient)
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Similar to Sorted Edge Algorithm, but no circuit.
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Minimim cost spanning tree == every vertex is connected to an other vertex
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Add cheapest up, unless it creates a circuit.
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