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The definition of an Eulerian path is a path in a graph which visits each edge exactly once. Intuitively, think of tracing the path with a pencil without lifting the pencil's edge from the page. One definition of an Eulerian graph is that every vertex has an even degree. You can check this by counting the degrees. Please see the related link for details.
The graph can be linear, if is just draw the line in the same path and see the Y axis answer. Sorry if you don't understand. =\
The longest path or the most indirect.
Simply, updating the existing distance with later received minimal value when a shortest path problem is solved in a graph. K.M.Anura Wijayasingha.
The answer depends on what variables the graph shows.