11.3 Vertex and edge covers

Vertex and edge covers are crucial concepts in graph theory. Vertex covers include vertices that touch every edge, while edge covers use edges to connect all vertices. These ideas help solve real-world problems like network security and facility location.

Finding minimum covers is a key challenge. For vertex covers, it's NP-hard in general but solvable for some graphs. Edge covers are easier, using maximum matching algorithms. These concepts link to other graph theory ideas and have wide-ranging applications.

Vertex and Edge Covers in Graphs

Vertex and edge covers

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• Vertex cover encompasses vertices including at least one endpoint of every edge covers all edges in graph (domino arrangement)
• Edge cover comprises edges covering all vertices every vertex incident to at least one edge in cover (spider web structure)
• Vertex covers emphasize vertices while edge covers prioritize edges vertex covers include one endpoint per edge edge covers connect to every vertex

Minimum cover determination

• Minimum vertex cover smallest set of vertices covering all edges NP-hard problem for general graphs polynomial-time algorithms for specific classes (bipartite graphs)
• Minimum edge cover smallest set of edges covering all vertices found in polynomial time using maximum matching algorithms
• Finding minimum covers utilizes:
  1. Greedy algorithms
  2. Approximation algorithms
  3. Integer linear programming formulations
• Minimum vertex cover complement yields maximum independent set (chessboard analogy)

Maximum matchings vs vertex covers

• Maximum matching largest set of edges without common endpoints (dance partners)
• König's theorem bipartite graphs maximum matching size equals minimum vertex cover size
• Gallai's theorem sum of minimum vertex cover and maximum matching sizes equals vertex count
• Maximum matching provides lower bound for minimum vertex cover size
• Algorithms leverage relationship to approximate minimum vertex covers construct covers from matchings in bipartite graphs

Applications of cover concepts

• Network security vertex covers determine minimum monitored nodes (security cameras)
• Facility location edge covers find optimal service placement covering all locations (pizza delivery)
• Task scheduling vertex covers assign minimum resources to cover all tasks (project management)
• Traffic control edge covers place traffic lights at minimum intersections (city planning)
• Optimization techniques employ:
  1. Linear programming relaxations
  2. Branch and bound algorithms
  3. Local search heuristics
• Real-world applications include sensor placement in IoT networks camera placement for surveillance systems protein-protein interaction networks in computational biology