Graphs and networks, the minimum spanning tree by Kruskal's and Prim's algorithms, the shortest path by Dijkstra's algorithm, and sorting and route-inspection ideas in decision mathematics.

Both algorithms find a minimum spanning tree (a set of edges connecting all vertices with the least total weight and no cycle).

Kruskal's algorithm is edge-based: list the edges in order of increasing weight, then add each edge in turn provided it does not create a cycle, stopping when all vertices are connected. It avoids a cycle by rejecting any edge whose two ends are already joined.

Prim's algorithm is vertex-based: start from any vertex and repeatedly add the shortest edge that connects a new vertex to the tree already built, until all vertices are included. It avoids a cycle by only ever adding an edge to a vertex not yet in the tree.

Markers reward the edge-based versus vertex-based distinction, the selection rule for each, and how each prevents a cycle.

Dijkstra's algorithm finds the shortest path from a start vertex to every other vertex in a network with non-negative weights.

Assign A a permanent label of $0$. Give each neighbour a temporary (working) label equal to the edge weight from A. Choose the vertex with the smallest temporary label and make it permanent. From that newly permanent vertex, update the temporary labels of its neighbours if a shorter route via it is found (the new label is the smaller of the current label and the permanent label plus the connecting edge).

Repeat, always making permanent the smallest temporary label, until E is permanent. The final permanent label of E is the length of the shortest path from A; trace the route backwards by following the edges that gave each label its final value.

Markers reward the permanent and temporary labelling, the update rule, the order of making labels permanent, and that the final label is the shortest distance with the route recovered by backtracking.