There are a few concepts within graph theory that include walks, paths paths, trails  and trails. cycles.  A walk on a graph is defined as a sequence of adjacent vertices where repetition is allowed. A path is a walk, however, no vertices can be repeated in this case. Notice that within these two concepts, it is known that if a walk exists between \(x\) and \(y\), then a path also exists from \(x\) to \(y.\) Next, a trial is defined as a walk that has no repeated edges and a trail may be closed if it starts and ends with the same vertex.