Tales by Dots and Lines
This Class 8 Ganita Prakash chapter introduces graphs in the sense of networks — dots joined by lines. Such graphs model roads between cities, friendships, and circuits. The chapter looks at vertices and edges, the degree of a vertex, paths and circuits, and the neat question of when a network can be drawn in a single stroke.
Learning objectives
- Describe a graph as vertices (dots) joined by edges (lines).
- Find the degree of a vertex.
- Use the rule that degrees add up to twice the number of edges.
- Decide when a network can be traced without lifting the pen.
Key concepts
Graphs as dots and lines
Here a graph is a network: dots called vertices joined by lines called edges. The dots can stand for places, people or junctions, and the lines for the connections between them — roads, friendships or wires. The picture matters only for which dots are joined, not for how the lines are drawn.
Degree of a vertex
The degree of a vertex is the number of edges meeting at it. A vertex with an even number of edges is an even vertex, and one with an odd number is an odd vertex. Degrees tell us a lot about how a network behaves.
The handshake idea
If we add up the degrees of all vertices, each edge is counted twice — once at each end. So the sum of all degrees equals twice the number of edges: Σ degrees = 2 × (number of edges). A useful consequence is that the number of odd-degree vertices is always even.
Paths, circuits and tracing
A path moves along edges from one vertex to another without repeating an edge; a circuit returns to where it started. A famous question asks when a network can be traced in one continuous stroke without lifting the pen or repeating an edge. This is possible exactly when the network has zero or two odd-degree vertices.
Important formulas
Degree sum (handshake)
sum of all degrees = 2 × number of edges
Traceable in one stroke
if the network has 0 or 2 odd-degree vertices
Key definitions
- Vertex
- A dot in a graph, representing a point such as a place or person.
- Edge
- A line joining two vertices, representing a connection.
- Degree
- The number of edges meeting at a vertex.
- Circuit
- A path that returns to its starting vertex without repeating an edge.
Solved examples
Q1. A graph has vertices of degrees 2, 3, 3 and 2. What is the total degree, and how many edges?
Solution: Total degree = 2+3+3+2 = 10, so edges = 10 ÷ 2 = 5.
Q2. How many odd-degree vertices does the graph above have?
Solution: Two vertices have odd degree (3 and 3), and two is an even count, as expected.
Q3. Can a network with exactly two odd vertices be traced in one stroke?
Solution: Yes — a network is traceable in one stroke when it has zero or two odd-degree vertices.
Common mistakes to avoid
- Thinking the way the lines are drawn (straight or curved) changes the graph — only the connections matter.
- Counting a vertex's degree wrongly when several edges meet there.
- Forgetting that each edge adds to two vertices' degrees.
- Believing every network can be traced in one stroke — it needs 0 or 2 odd vertices.
Tales by Dots and Lines — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
In a graph, the dots are called:
Practice questions
Short answer
What is the degree of a vertex?
The number of edges meeting at that vertex.
State the handshake rule for graphs.
The sum of all vertex degrees equals twice the number of edges.
When can a network be traced in a single stroke?
When it has zero or two vertices of odd degree.
Long answer
Explain why the sum of the degrees of a graph is always twice the number of edges.
Every edge joins exactly two vertices, so when we count the degree at each vertex, that edge is counted once at each of its two ends — a total of two. Adding the degrees of all vertices therefore counts every edge twice, giving sum of degrees = 2 × number of edges. A direct consequence is that the number of odd-degree vertices must be even, since the total of all degrees is an even number.
Describe the rule for tracing a network in one stroke, with an example.
A connected network can be drawn in one continuous stroke, without lifting the pen or repeating an edge, exactly when it has either zero or two vertices of odd degree. If there are no odd vertices, the trace can start anywhere and returns to the start; if there are two, the trace must start at one odd vertex and finish at the other. For example, a single square (each corner has degree 2, all even) can be traced in one stroke, but a network with four odd vertices cannot.
HOTS (Higher Order Thinking)
A network has degrees 4, 4, 3 and 1. Can it be traced in one stroke? Why?
It has exactly two odd-degree vertices (3 and 1), so yes — it can be traced in one stroke, starting at one odd vertex and ending at the other.
Why is it impossible for a graph to have exactly three odd-degree vertices?
Because the number of odd-degree vertices must be even (the degrees sum to twice the edges), so three is impossible.
Quick revision
Revision notes
- Graph = vertices (dots) joined by edges (lines); only connections matter.
- Degree = edges at a vertex; even or odd vertex.
- Sum of degrees = 2 × edges; odd vertices come in even numbers.
- Traceable in one stroke ⇔ 0 or 2 odd vertices.
Key takeaways
- A graph models connections, not shapes.
- Each edge contributes to two degrees.
- Odd vertices decide whether a network can be drawn in one stroke.
Frequently asked questions
What is a graph in this chapter?
A network of dots (vertices) joined by lines (edges) showing connections.
What does the degree of a vertex mean?
How many edges meet at that vertex.
How do I know if a network can be traced in one stroke?
It can if it has zero or exactly two odd-degree vertices.