Subgraphs are essential building blocks in graph theory, allowing us to analyze smaller parts of larger networks. They're formed by taking subsets of vertices and edges from a bigger graph, giving us tools to study complex structures piece by piece.
Understanding subgraphs opens doors to powerful graph operations like union, intersection, and complement. These operations let us combine, compare, and manipulate graphs in various ways, helping us model real-world systems and solve intricate problems in network analysis.
Understanding Subgraphs
Concept of subgraphs
- Subgraph forms from subset of vertices and edges of another graph
- Types include induced subgraph (contains all edges between its vertices present in original graph) and spanning subgraph (includes all vertices of original graph)
- Subgraphs have fewer or equal vertices and edges compared to original graph inherit properties from parent graph
- Notation $H \subseteq G$ denotes H is a subgraph of G
Identification of subgraphs
- Methods involve vertex deletion and edge deletion
- Process: select subset of vertices from original graph include connecting edges verify resulting structure forms valid subgraph
- Special subgraphs: cliques (complete subgraphs), paths (sequences of connected vertices), cycles (closed paths)
- Subgraph identification crucial in network analysis and pattern recognition (social networks, molecular structures)
Graph operations
- Graph union $G_1 \cup G_2$ combines vertices and edges of two graphs
- Graph intersection $G_1 \cap G_2$ retains only common vertices and edges between two graphs
- Graph complement $\overline{G}$ or $G^c$ creates new graph with edges where original had none
- Edge operations include deletion (removing edges) and contraction (merging vertices)
- Sum of edges in graph and its complement equals total possible edges
- Double complement returns original graph
Creation through graph operations
- Techniques involve combining multiple existing graphs or modifying single graph through operations
- Process: identify desired properties select appropriate operations apply in specific order to achieve result
- Examples: constructing complete bipartite graph from two independent sets creating wheel graph by adding central vertex to cycle graph
- Applications in modeling complex systems (transportation networks) and generating test cases for graph algorithms
- Verify newly created graphs by checking properties like connectivity planarity or regularity