Consider the following List-I and List-II in context of graph & model.

	List – I		List – II
A.	Planar Graph	I.	Probabilistic Model
B.	Bipartite Graph	II.	Deterministic Model
C.	PERT	III.	4 - Colorable
D.	CPM (Critical Path Method)	IV.	2 - Colorable

Choose the correct answer from the options given below:

Correct option is A

We need to match the graph-theoretic concepts and project-management models in List-I with their corresponding descriptions in List-II.
A. Planar Graph → III. 4-Colorable
A planar graph is a graph that can be drawn on a plane without any two edges crossing except at their common vertices.
The Four-Color Theorem states that:
Every planar graph can be colored using at most four colors so that no two adjacent vertices have the same color.
B. Bipartite Graph → IV. 2-Colorable
A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.
Therefore, its vertices can be colored using two colors, with adjacent vertices receiving different colors.
C. PERT → I. Probabilistic Model
PERT (Program Evaluation and Review Technique) is used for project planning and scheduling when activity durations are uncertain.
PERT uses three-time estimates:
· Optimistic time $$(t_o)$$​​
· Most likely time $$(t_m)$$​​
· Pessimistic time $$(t_p)$$​​
The expected activity time is:
​$$t_e=\frac{t_o+4t_m+t_p}{6}$$​​
Because PERT incorporates uncertainty in activity durations, it is considered a probabilistic approach/model.
D. CPM → II. Deterministic Model
CPM (Critical Path Method) assumes that activity durations are known or can be estimated with reasonable certainty.
It uses a single time estimate for each activity and determines the critical path—the longest-duration path through the project network.
Therefore, CPM is considered a deterministic model.
Information Booster
1. Planar graph → 4-colorable (Four Color Theorem).
2. Bipartite graph → 2-colorable.
3. A graph is bipartite if and only if it contains no odd cycle.
4. PERT uses three time estimates and is probabilistic.
5. CPM generally uses a single deterministic time estimate.
6. PERT is particularly useful when activity durations are uncertain, while CPM is commonly used when durations are relatively predictable.