Multiple paths of same minimum weight

7 [Figure 2, printed on the insert, is provided for use in this question.]
The following network has eight vertices, \(A , B , \ldots , H\), and edges connecting some pairs of vertices. The number on each edge is its weight. The weights on the edges \(E H\) and \(G H\) are functions of \(x\) and \(y\). \includegraphics[max width=\textwidth, alt={}, center]{4c5c963b-0183-4dc7-9054-b2c7a3eb8c1b-07_1170_1705_596_164} Given that there are three routes from \(A\) to \(H\) with the same minimum weight, use Dijkstra's algorithm on Figure 2 to find:

1. this minimum weight;
2. the values of \(x\) and \(y\).

7 [Figure 2, printed on the insert, is provided for use in this question.]
The following network has 13 vertices and 24 edges connecting some pairs of vertices. The number on each edge is its weight. The weights on the edges \(G K\) and \(L M\) are functions of \(x\) and \(y\), where \(x > 0 , y > 0\) and \(10 < x + y < 27\). \includegraphics[max width=\textwidth, alt={}, center]{f99fad35-3304-4e8f-be02-1439dfdc10e1-7_1218_1431_660_312} There are three routes from \(A\) to \(M\) of the same minimum total weight.

1. Use Dijkstra's algorithm on Figure 2 to find this minimum total weight.
2. Find the values of \(x\) and \(y\).

\includegraphics{figure_1} **Figure 1** [The total weight of the network is \(135 + 4x + 2y\)] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of \(x\) and \(y\), where \(x\) and \(y\) are positive constants and \(7 < x + y < 20\) There are three paths from A to H that have the same minimum length.

1. Use Dijkstra's algorithm to find \(x\) and \(y\). [7]

An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length.

1. State the arcs that are traversed twice. [1]
2. State the number of times that vertex C appears in the inspection route. [1]
3. Determine the length of the inspection route. [1]