Math Problem Statement
What is the size of the minimum vertex cover of graph G?
Solution
To determine the size of the minimum vertex cover of the graph represented by the given adjacency matrix, let’s proceed step by step:
Step 1: Understand the problem
- Adjacency Matrix: The matrix provides information about the connections (edges) between vertices in an undirected graph .
- A vertex cover is a subset of vertices such that every edge of the graph is incident to at least one vertex in the subset.
- The minimum vertex cover is the smallest such subset.
Step 2: Analyze the matrix
The given adjacency matrix is:
0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \end{bmatrix}$$ From this matrix: 1. The graph has **5 vertices**. 2. Each entry $$A[i][j] = 1$$ (off-diagonal) indicates an edge between vertex $$i$$ and $$j$$. ### Step 3: Construct the graph (describe edges) The edges of the graph are: - From $$v_1$$: $$v_1 \to v_2, v_1 \to v_4$$ - From $$v_2$$: $$v_2 \to v_1, v_2 \to v_3, v_2 \to v_5$$ - From $$v_3$$: $$v_3 \to v_2, v_3 \to v_4, v_3 \to v_5$$ - From $$v_4$$: $$v_4 \to v_1, v_4 \to v_3, v_4 \to v_5$$ - From $$v_5$$: $$v_5 \to v_2, v_5 \to v_3, v_5 \to v_4$$ ### Step 4: Find the minimum vertex cover - A **vertex cover** ensures all edges are covered by at least one endpoint. - A **minimum vertex cover** can be computed by applying methods like bipartite matching (if the graph is bipartite) or approximation algorithms for general graphs. #### Observations: - The graph is dense, with each vertex having multiple connections. - Using a greedy method, we can iteratively pick vertices with the highest degree to cover edges. #### Solution: By inspection: - Select vertices $$v_2, v_3, v_4$$. This subset covers all edges: - $$v_2$$: Covers $$v_1 \to v_2, v_2 \to v_3, v_2 \to v_5$$ - $$v_3$$: Covers $$v_3 \to v_4, v_3 \to v_5$$ - $$v_4$$: Covers $$v_1 \to v_4, v_4 \to v_5$$ Thus, the **minimum vertex cover** has a size of **3**. ### Final Answer: The size of the minimum vertex cover is **3**. --- Would you like a detailed explanation of how vertex covers relate to other graph-theory concepts? Here are 5 related questions: 1. What is the relationship between the size of the minimum vertex cover and the maximum matching in a graph? 2. How can you find the minimum vertex cover using a formal algorithm (e.g., linear programming)? 3. Is the graph bipartite? How would that affect the minimum vertex cover? 4. How does the adjacency matrix help in identifying cliques or independent sets? 5. What is the computational complexity of finding the minimum vertex cover in a general graph? **Tip**: For dense graphs, approximation algorithms like a 2-approximation for vertex cover are often efficient!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Graph Theory
Vertex Cover
Adjacency Matrix
Formulas
-
Theorems
Vertex Cover Theorem
Relationship between Vertex Cover and Maximum Matching
Suitable Grade Level
Undergraduate (College level)
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