5.7 Eulerizing graphs
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The objective:
- turn it into a graph all of whose vertices have even degree
- by adding some new edges which are "duplicates" of currently existing edges:
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this eulerized graph now has an Euler circuit!
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Note that all 4 vertices are odd, so we will duplicate edges RD and AL,
thus making all vertices even!
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- of course, such a tour is a compromise between
- one the one hand, no tour and
- on the other, building new bridges
- and only approximates a real Euler circuit . . .
- but it's the best we can do for the poor folks of Konigsberg!
- remember:
- added edges must be duplicates of existing edges
- we're not asking that the Konigsbergers actually build new bridges
- we're just allowing some bridges to be recrossed
- and this rule is not just for Konigsberg; it holds for all eulerizations
- so here's a no-no; it is NOT an eulerization:
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Optimal
eulerizations
- if the objective is to construct an Euler circuit that traverses as few edges as possible
- we want to add as few edges as possible
- adding as few edges as possible is called an optimal eulerization
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Semi-eulerizations
If all we need is an Euler path (not a circuit), we can leave two of the vertices odd, which will then be the start and end of an Euler path. This is called a semi-eulerization:
If all we need is an Euler path (not a circuit), we can leave two of the vertices odd, which will then be the start and end of an Euler path. This is called a semi-eulerization:
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