Chapter 5: Euler Paths and Circuits (MATH 118)
What is the Sum of Degrees Theorem?
1) A graph always has an even number of vertices with odd degrees. 2) The sum of all the degrees of a graph equals twice the number of edges, and will always be an even number (duh).
What are the steps to Fleury's Algorithm of Euler Paths and Circuits?
1) Check correctness (no odd vertices or exactly odd vertices). 2) Choose starting vertex (has to be odd if there are 2 odd vertices). 3) If you have a choice, avoid a bridge. If you don't have a choice, take the bridge. 4) If you can't travel any more, you're finished.
What is the Euler Circuit Theorem?
1) If a graph has all even vertices, then it has a Euler circuit. 2) If a graph has any odd vertices, then it doesn't have a Euler circuit. 3) If a graph has exactly 2 odd vertices, the is has a Euler path.
What is the Euler Path Theorem?
1) If a graph has more than 2 odd vertices, it doesn't have a Euler path. 2) If a graph has exactly 2 vertices, it has a Euler path. 3) You must start and end on one of the odd vertices (but not always the same one).
What is a Euler circuit?
A Euler path that starts and ends on the same vertex.
What is a circuit?
A closed trip on a graph where the starting and closing vertices are the same.
What is a unicursal tracing?
A graph that can be drawn without picking up you pen and without retracing any part of the graph.
What is an algorithm?
A procedure that leads to a "solution." (It doesn't always find "the best" solution, but one that works.)
What is a trip?
A sequence of adjacent edges with the property than an edge can only be traveled once.
What is a graph?
A structure that consists of a set of objects and a list describing how pairs of objects are related.
What is a even vertex?
A vertex with an even number of edges connected to it.
What is a odd vertex?
A vertex with an odd number of edges connected to it.
What is a bridge?
An edge that can be deleted to make a connected graph a disconnected one.
What does "bridge" mean in the context of Fleury's Algorithm?
An edges that becomes a bridge after deleting already-traversed edges.
What is a vertex?
An object within a graph.
What is a path?
An open trip on a graph where the starting and closing vertices are different.
If a graph has 4, 6, 8... odd vertices...
It doesn't have a Euler path or circuit.
If a graph has 0 odd vertices...
It has a Euler circuit.
If a graph has 2 odd vertices...
It has a Euler path.
If a graph has 1, 3, 5... odd vertices...
It isn't a possible graph.
What are edges?
Relations between each pair of objects.
What is connectedness?
The idea that you can reach any vertex of a graph by traversing the edges given in the graph.
What is the length of a path?
The number of edges in a path.
What is the degree of a vertex?
The number of edges touching that vertex.
How is a graph semi-Eulerized?
When an edge is "added" so there are exactly 2 odd vertices, making a Euler path.
How is a graph Eulerized?
When an edge is "added" to odd vertices to make an Euler circuit possible. Only already existing edges can be duplicated, no adding new edges.
When are graphs equivalent?
When both graphs have the same relations, meaning that the same edges connect the same vertices.
What is a disconnected graph?
When one or more vertices of a graph isn't connected to the rest of the graph.
What is a "clique"?
Where every vertex in a graph is related to every other vertex in the graph.