**Duelling Connections**Consider a graph which represents the flights made by two airlines among $ N $ cities. Between every pair of cities, exactly one airline flies the route. Show that there exists an airline which may carry a passenger from any city to any other city using only their own routes.For example, consider the particular graph:The graph includes four nodes labeled as City 1, City 2, City 3, and City 4. These nodes are connected by edges of two different colors representing two airlines: Airline 1 (blue) and Airline 2 (red). Each pair of cities is connected by one colored route corresponding to one of the airlines.- Airline 1's routes are depicted with blue edges.- Airline 2's routes are depicted with red edges.Airline 1 is able to carry passengers from any city to any other using only their routes. Notice that to achieve this, Airline 1 may use indirect flights.The graph illustrates how each airline covers all cities, with each route marked clearly. The indirect flights mean that Airline 1 can still connect any two cities even without a direct route between them.

We will try to answer the question using the figure given in the example and taking the case of…

Duelling Connections Consider a graph which represents the flights made by two airlines among N cities. Between every pair of cities exactly one airline flies the route. Show that there exists an airline which may carry a passenger from any city to any other city using only their own routes. For example, consider the particular graph: CTYI CITYI ARLINE a Airline 1 is able to carry passengers from any city to any other using only their routes. Notice that to achieve this Airline 1 may use indirect flights.

Duelling Connections Consider a graph which represents the flights made by two airlines among N cities. Between every pair of cities exactly one airline flies the route. Show that there exists an airline which may carry a passenger from any city to any other city using only their own routes. For example, consider the particular graph: CTYI CITYI ARLINE a Airline 1 is able to carry passengers from any city to any other using only their routes. Notice that to achieve this Airline 1 may use indirect flights.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Duelling Connections**

Consider a graph which represents the flights made by two airlines among $ N $ cities. Between every pair of cities, exactly one airline flies the route. Show that there exists an airline which may carry a passenger from any city to any other city using only their own routes.

For example, consider the particular graph:

The graph includes four nodes labeled as City 1, City 2, City 3, and City 4. These nodes are connected by edges of two different colors representing two airlines: Airline 1 (blue) and Airline 2 (red). Each pair of cities is connected by one colored route corresponding to one of the airlines.

- Airline 1's routes are depicted with blue edges.
- Airline 2's routes are depicted with red edges.

Airline 1 is able to carry passengers from any city to any other using only their routes. Notice that to achieve this, Airline 1 may use indirect flights.

The graph illustrates how each airline covers all cities, with each route marked clearly. The indirect flights mean that Airline 1 can still connect any two cities even without a direct route between them.

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