Question

a city is built on the banks of a river and some islands in the river. the map below shows the bridges connecting the various land masses. draw a graph that models the connecting relationships in the map below. the vertices represent the land masses and the edges represent bridges connecting them.
is it possible to find a circuit through the city that uses each bridge once? if so, enter the sequence of land masses(vertices) visited, for example abdea. if it is not possible, enter dne.

Explanation:

Step1: Identify Vertices and Degrees

Vertices: \( A, B, C, D, E \).

• \( A \): connected to \( C, D, E \) → degree \( 3 \).
• \( B \): connected to \( C, D, E \) → degree \( 3 \).
• \( C \): connected to \( A, B \) → degree \( 2 \).
• \( D \): connected to \( A, B \) → degree \( 2 \).
• \( E \): connected to \( A, B \) → degree \( 4 \).

Eulerian circuit exists if all vertices have even degrees. Here, \( A, B \) have odd degrees (\( 3 \)), so no Eulerian circuit.

Step2: Apply Eulerian Circuit Theorem

A graph has an Eulerian circuit iff all vertices have even degrees. Since \( A \) and \( B \) have odd degrees, no such circuit.

Answer:

DNE