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Any connected group of things can be represented as a graph: cities and roads, people and friendships, and more. Learn why an even number of people have an odd number of friends.

Cosmic liaisons are established among the nine planets of the solar system. Rockets can only travel along the following routes:

Can a traveler get from Earth to Mars?
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Five cities \(P,Q,R,S,T\) are connected by different modes of transport as follows

\(P\) and \(Q\) connected by boat as well as rail.

\(S\) and \(R\) connected by bus and boat.

\(Q\) and \(T\) connected by air only.

\(P\) and \(R\) connected by boat only.

\(T\) and \(R\) connected by rail and bus.

If a person visits each of the places starting from \(P\) and gets back to \(P,\) which of the following places must he visit **twice**?

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Here is a network graph constructed with data from Facebook of 20 people and all of the mutual friendship connections among them. Clearly Barbie has the most friends to invite to her parties (9), but if invitations go out not only to friends but *also* to all friends of friends, then whose party will have the most invites?

**Notes and assumptions**

- Each dot represents a person.
- Each line represents mutual friendship between the people on either end .

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**cannot** be traced without lifting up your pencil and without tracing over the same edge twice?

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