Sample algorithm to "linearize" a graph -

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simplifying business example, have following situation:

some objects should distributed in graph in "linear" way possible given "thermometer".

say, voyager visits cities. several cities visited multiple times.

so, have list of cities in ordinate axis (that may duplicated), , time in abscissas one.

now, given path, (a => x => => b => c) should display line, in "most linear way possible".

enter image description here

by eg. in image above, green line optimal one
(1 > 2 > 3 > 4 > 5)

but there multiple possible outputs

(1 > 2 > 1 > 4 > 5)
(1 > 2 > 3 > 4 > 5)
(1 > 2 > 6 > 4 > 5)

(3 > 2 > 1 > 4 > 5)
(3 > 2 > 3 > 4 > 5)
(3 > 2 > 6 > 4 > 5)

(6 > 2 > 1 > 4 > 5)
(6 > 2 > 3 > 4 > 5)
(6 > 2 > 6 > 4 > 5)

is there algorithms helping in such situations?

construct graph node pairing of city+value , time (e.g. a(3)/1). edge exists between 2 nodes adjacent in path (e.g. a(3)/1 x(2)/2).

the weight of edge difference in vector (the opposite angle) between last pair of nodes , next pair of nodes (this make edge weight dynamic depending on came from). use dijkstra find minimum distance (a) goal.

example graph (edges given in degrees , estimates): 

                                            total cost       0        0      105       15 a31  ->  x22  ->  a13  ->  b44  ->  c55     120                90       0        0               ->  a33  ->  b44  ->  c55     90                115      110      105               ->  a63  ->  b44  ->  c55     330

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