WORKED EXAMPLE 1.2
Using the graph in Worked example 1.1:
a give an example of a walk that passes through every vertex b give two examples of trails, one with a repeated vertex and one with no repeated vertices c explain why A − B − C
− D
is not a trail
d give an example of a cycle e explain why E − B − A − C
− A − B − E
b An example of a trail with a repeated vertex is A − B − C − A − D
.
An example of a trail with no repeated vertices is D − A − B − E. c There is no edge C
is not a cycle. This is not a trail because the edge A − B is repeated.
PL E
a e.g. D − A − B − C
− B − E
− D
This trail repeats the vertex A.
This trail has no repeated vertices.
.
− A
M
d For example: A − B − C
e Vertex B is travelled through twice.
A cycle has no repeated vertices, apart from starting and finishing at the same vertex.
SA
Key point 1.3
Two vertices are directly connected, or adjacent, if there is an edge with these vertices at its ends. An indirect connection between two vertices passes through other vertices and involves more than one edge. A graph is connected if it is possible to get from any vertex to any other, directly or indirectly.
The position of the vertices and the shapes of the edges in a graph (including whether they cross each other or not) are irrelevant. All that matters is which vertices are adjacent (directly joined) to each other.
Key point 1.4 An edge that directly connects a vertex to itself is called a loop. A graph has a multiple edge if there are two or more edges that directly connect the same pair of vertices. Original material © Cambridge University Press 2018