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Chapter 3: Q30E (page 112)

On page 102, we defined the binary relation 鈥渃onnected鈥� on the set of vertices of a directedgraph. Show that this is an equivalence relation(see Exercise 3.29), and conclude that it partitions the vertices into disjoint strongly connected components.

Short Answer

Expert verified

It can be shown that the binary relation 鈥渃onnected鈥� on the set of vertices of a directed graph is an equivalence relation and yes, it partitions the vertices into disjoint strongly connected components.

Step by step solution

01

Explain the Equivalence relation

A relation is said to be in equivalence only if the relation satisfies reflexive, symmetry, and transitive properties.

02

Show that the given relation is the equivalence relation

Consider a set S that has the partitions of an undirected graph. Consider any two vertices x and y in the undirected graph.

From the solution of Exercise 3.29, the binary connected relation of the connected relationship satisfies reflexivity, symmetry, and transitivity. So, it is an equivalence relation.

The strongly connected component is the equivalence class corresponding to this relation.

Thus, it partitions the vertices into disjoint strongly connected components.

Therefore, It is shown that the binary relation 鈥渃onnected鈥� on the set of vertices of a directed graph is an equivalence relation and yes, it partitions the vertices into disjoint strongly connected components.

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Most popular questions from this chapter

Either prove or give a counterexample: if {u,v}is an edge in an undirected graph, and during depth-first search (u)<post (v), then vis an ancestor of uin the DFS tree.

Rewrite the explore procedure (Figure 3.3) so that it is non-recursive (that is, explicitly use a stack). The calls to pre visit and post visit should be positioned so that they have the same effect as in the recursive procedure.

Give a linear-time algorithm to find an odd-length cycle in a directed graph. (Hint: First solve this problem under the assumption that the graph is strongly connected.)

A bipartite graph is a graph G=(V,E)whose vertices can be partitioned into two sets (V=V1V2andV1V2=蠒) such that there are no edges between vertices in the same set (for instance, if , then there is no edge between and ).

(a) Give a linear-time algorithm to determine whether an undirected graph is bipartite.

(b) There are many other ways to formulate this property. For instance, an undirected graph is bipartite if and only if it can be colored with just two colors. Prove the following formulation:

an undirected graph is bipartite if and only if it contains no cycles of odd length.

(c) At most how many colors are needed to color in an undirected graph with exactly one odd length?

Give a linear-time algorithm for the following task.
Input: A directed acyclic graph G

Does G contain a directed path that touches every vertex exactly once?
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