Dots and lines: networks and puzzles

A network has one centre dot joined by lines to 5 outer dots, and no other lines. What is the degree of the centre dot?

The dots of a network have degrees 3, 3, 2, 4 and 2. How many lines does the network have?

Amaru says he has drawn a network whose four dots have degrees 3, 2, 2 and 2. Is that possible?

Each figure below is connected, and the numbers are the degrees of its dots. Which figures can be traced in one stroke? Select all that apply.

A connected figure has five dots with these degrees: P has 2, Q has 3, R has 4, S has 3 and T has 2. Where must a one-stroke tracing start?

Ama walks from her house to the swimming pool. The roads are: house–library 4 km, library–pool 5 km, house–park 3 km, park–pool 7 km, park–shop 3 km, shop–pool 2 km. What is the shortest distance from her house to the pool, in kilometres?

The people of Königsberg decide to build new bridges so that a walk crossing every bridge exactly once becomes possible. (The walk may start and finish in different places.) What is the smallest number of new bridges they need?

Lean has checked the theorem odd_dot_is_an_end: if a route traces a network in one stroke, then every odd dot is the start or the end of the route. Which statement can Gideon conclude from this theorem alone?

Margaret writes a network as a list of lines, each line a string of two dot letters, like ["AB", "BC", "CA"] for a triangle. Complete the function degree(lines, dot) so that it returns the number of line ends at dot. Two lines may join the same two dots (like bridges).