Let's analyze the given graph to find the requested properties. Here's what the problem entails:

(a) All cycles:

Cycles are closed paths in the graph where vertices are visited exactly once except for the starting/ending vertex.

(b) All cut points:

A cut point (or articulation point) is a vertex that, when removed along with its incident edges, increases the number of connected components in the graph.

(c) All bridges:

A bridge (or cut edge) is an edge that, when removed, increases the number of connected components in the graph.

I'll analyze this graph based on its structure:

Step 1: Analyze the structure

The graph is composed of multiple connected components:

1. $A$-$B$-$J$-$K$ forms a subgraph with a triangle (cycle).
2. $C$-$D$-$L$-$M$ appears linear with no triangles.

Solution

(a) All Cycles

From the graph:

• The only cycle is $A$-$B$-$K$-$J$-$A$ (a quadrilateral).
• No additional cycles exist.

(b) All Cut Points

A cut point disconnects the graph when removed. From the graph:

• Vertex $B$ is a cut point, as its removal separates $A, J, K$ from the rest of the graph.
• Vertex $L$ is a cut point, as its removal separates $C, D$ from $M$.

(c) All Bridges

A bridge disconnects the graph into two components when removed. From the graph:

• The edges $B$-$L$, $L$-$C$, $L$-$M$, and $C$-$D$ are bridges.

Would you like more detailed steps for any part of the solution? Here are some relative questions for further exploration:

1. What is the mathematical definition of a cycle, cut point, and bridge in a graph?
2. How would you apply depth-first search (DFS) to find all cut points and bridges in a graph?
3. Can this graph be represented in adjacency matrix form? If so, how would it look?
4. How do cycles in planar graphs like this relate to Eulerian paths or circuits?
5. What would change in the analysis if we added or removed edges from this graph?

Tip: When analyzing cut points and bridges, consider systematically removing each vertex or edge and observing the connected components to verify your results!