+Section 1.7 Defining Functions
-The graph \(G' = (V', E')\) is a subgraph of \(G = (V, E)\) if \(V' \subseteq V\) and \(E' \subseteq \{\{u, v\} \in E \ |\ u, v \in V'\}\text{.}\)
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-The subgraph \(G' = (V', E')\) is a spanning subgraph of \(G = (V, E)\) if \(V' = V\text{.}\)
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-A spanning tree for the graph \(G\) is a spanning subgraph of \(G\) that is a tree.
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Breadth-first search algorithm
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-Choose a vertex of the graph (root), arbitrarily.
Travel all the edges incident with the root vertex.
Give an order to this set of new vertices added.
Consider each of these vertices as a root, in order, and add all the unvisited incident edges that do not produce a cycle.
Repeat the method with the new set of vertices.
Follow the same procedure until all the vertices have been visited.
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The output of this algorithm is a spanning tree.
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Given a weighted graph, a minimal spanning tree is a spanning tree whose sum of wights is minimal.
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-Prim’s Algorithm: Keep two disjoint sets of vertices. One \((L)\) contains vertices that are in the growwing spanning tree and the other \((R)\) that are not in the growing spanning tree.
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-Choose a vertex \(u \in V\) arbitrarily. At this step, \(L = \{u\}\) and \(R = V - \{u\}\text{.}\)
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Select the cheapest vertex in \(R\) that is connected to the growing spanning tree \(L\) and add it into \(L\)
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Follow the same procedure until all the vertices are in \(L\)
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The output of this algorithm is a minimal spanning tree.
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Let’s explore the following graph:
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+You can also add input validation to your functions to make them more robust.
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+Try calling the function with an empty string or a non-string argument to see the input validation in action.
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-We can ask Sage for the minimal spanning tree of this graph.
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-Let’s visualize the minimal spanning tree.
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