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Chapter 11: Q3E (page 795)

In Exercises 2–6 find a spanning tree for the graph shown by removing edges in simple circuits.

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01

Compare with the definition.

A spanning tree of a simple graph G is a subgraph of G that is a tree and that contains all vertices of G.

A tree is an undirected graph that is connected and that does not contains any single circuit. And a tree with n vertices has n-1 edges.

Here graph contains 7 vertices and 14 edges.

The spanning tree contains 7 vertices and 6 edges. Thus8 edges will be remove from the graph.

02

Remove the extra edges.

Since a tree cannot contains any single circuit and there are two tangents present in the graph. Then suffices to remove one edge from either triangle. Thus removing the edges (a,g),(d,g),(d,f),(b,e),(c,e).

Since all curve edges in the remaining graph still create a circuit.so remove other edges are (a,d),(d,g),(e,g).

Now. Thespanning tree will be as

This is the required result.

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Most popular questions from this chapter

In this exercise we will develop an algorithm to find the strong components of a directed graph \({\bf{G = }}\left( {{\bf{V,E}}} \right)\). Recall that a vertex \({\bf{w}} \in {\bf{V}}\) is reachable from a vertex \({\bf{v}} \in {\bf{V}}\) if there is a directed path from v to w.

  1. Explain how to use breadth-first search in the directed graph G to find all the vertices reachable from a vertex \({\bf{v}} \in {\bf{G}}\).
  2. Explain how to use breadth-first search in \({{\bf{G}}^{{\bf{conv}}}}\) to find all the vertices from which a vertex \({\bf{v}} \in {\bf{G}}\) is reachable. (Recall that \({{\bf{G}}^{{\bf{conv}}}}\) is the directed graph obtained from G by reversing the direction of all its edges.)
  3. Explain how to use part (a) and (b) to construct an algorithm that finds the strong components of a directed graph G, and explain why your algorithm is correct.

Show that a directed graph \({\bf{G = }}\left( {{\bf{V,E}}} \right)\) has an arborescence rooted at the vertex r if and only if for every vertex \({\bf{v}} \in {\bf{V}}\), there is a directed path from r to v.

Is the rooted tree in Exercise \(3\) a full \({\bf{m}}\)-ary tree for some positive integer \({\bf{m}}\)?

Can there be two different simple paths between the vertices of a tree?

Draw a game tree for him if the starting position consists of two piles with two and three stones, respectively. When drawing the tree represent by the same vertex symmetric positions that result from the same move. Find the valueof each vertex of the game tree. Who wins the game if both players follow an optimal strategy?
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