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Chapter 0: Q8E (page 26)

Consider the undirected graph G=(V,E)whereV, the set of nodes, is{1,2,3,4}
andE, the set of edges, is{{1,2},{2,3},{1,3},{2,4},{1,4}}. Draw the graphG. What are the degrees of each node? Indicate a path from node 3 to node 4 on your drawing ofG.

Short Answer

Expert verified

Below is the Graph G,

The degree of each node is as follows:

  • Node 1 has degree 3.
  • Node 2 has degree 3.
  • Node 3 has degree 2.
  • Node 4 has degree 2.

The path from node 3 to node 4 is shown below using the blue color:

Step by step solution

01

Describe the steps taken to draw the graph.

  1. Draw four circles as the nodes of the graph and number them.
  2. Connect nodes with a line for which there is an ordered pair in set E. For instance, (1,2) is given in the set, so make node 1 and node 2 connected.
02

Describe the degree of each node

A degree can be defined as the number of incoming or outgoing edges to a node.

  • There are 3 edges directly linked to nodes 1 and 2, so their degree is 3.
  • There are 2 edges directly linked to nodes 3 and 4, so their degree is 2.
03

Describe the path from node 3 to node 4

A path can be defined as the set of connected edges through which we can reach from one node to another node.

There can be many possible paths from node 3 to node 4, some are as follows:

3鈫�2鈫�43鈫�1鈫�43鈫�1鈫�2鈫�43鈫�2鈫�1鈫�4

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Most popular questions from this chapter

Show that A is decidable iff A鈮�m0*1*.

Let F be the language of all strings over 0,1that do not contain a pair of 1s that are separated by an odd number of symbols. Give the state diagram of a DFA with five states that recognizes F . (You may find it helpful first to find a 4-state NFA for the complement of F).

This problem is inspired by the single-player game Minesweeper, generalized to an arbitrary graph. Let Gbe an undirected graph, where each node either contains a single, hidden mine or is empty. The player chooses nodes, one by one. If the player chooses a node containing a mine, the player loses. If the player chooses an empty node, the player learns the number of neighboring nodes containing mines. (A neighboring node is one connected to the chosen node by an edge.) The player wins if and when all empty nodes have been so chosen.

In the mine consistency problem, you are given a graphG along with numbers labeling some of G鈥檚 nodes. You must determine whether a placement of mines on the remaining nodes is possible, so that any node v that is labeled m has exactly m neighboring nodes containing mines. Formulate this problem as a language and show that it isNPcomplete.

Let D=w|wcontains an even number of 鈥檚 and an odd number of 鈥檚 and does not contain the substring ab}. Give a DFA with five states that recognizes Dand a regular expression that generates D.(Suggestion: Describe Dmore simply.)

Give a counterexample to show that the following construction fails to prove Theorem 1.49, the closure of the class of regular languages under the star operationLet N1=Q1,危,未1,q1,F1 recognize . Construct N=Q1,危,未,q1,F as follows. Nis supposed to recognize A*1.

a. The states of Nare the states of N1.

b. The start state ofN is the same as the start state ofN1 .

c. . F=q1鈭�F1.The accept states are the old accept states plus its start state.

d. Define未so that for any and any a鈭�危蔚, 未q,a=(未1q,aq6鈭�F1ora6=蔚未1q,a鈭�q1q鈭�F1anda=蔚
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