Cycles
المؤلف:
Jean-Claude Fournier
المصدر:
Graph Theory and Applications
الجزء والصفحة:
31-33
2-8-2016
2028
A walk, a trail or a path (x0,e1,x1,...,ek, xk) is said to be closed if its ends x0 and xk coincide.
A cycle is a closed path of length ≥ 1, that is a path of the form:
where k ≥ 1, and vertices xi, for i =0,...,k − 1, are all distinct. Integer k is the length of the cycle. Unlike a walk, and a trail, a cycle cannot have
zero length. The minimum length it can be is 1. In this case, it is made up of one vertex with a loop. When G is a simple graph, a cycle may be defined
by the sequence (x0,x1,...,x0) of its vertices. In this case the length is then umber of vertices of the cycle (except the last one). A cycle is called even
or odd, depending on whether its length is even or odd.
Figure 1.1 gives some examples of walks, trails, paths and cycles.
Figure 1.1
Note. No distinction is made between cycles that only differ in the cyclic sequences of vertices which define them. For example, in a simple graph, the
three following cycles are considered as one unique cycle:
In fact, in the graph, it is the same cycle described differently.
Graph Theory and Applications ,Jean-Claude Fournier, WILEY, page(31-33)
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