Introduction to Graphs

For each, what are the vertices and what are the edges?

• Web pages with links
  • pages are vertices, embedded links the edges
• Methods in a program that call each other
  • methods are vertices, calls are edges
• Road maps (e.g., Google maps)
  • locations are vertices, streets/highways are edges
• Airline routes
  • locations are vertices, flights are edges
• Family trees
  • people are vertices, relationships are edges
• Course pre-requisites
  • courses are vertices, pre-reqs are edges

• Can think of graphs as an ADT with operations like isEdge(v1, v2) to check if there's an edge between two vertexes
• But it is unclear what the standard operations are
• Instead we tend to develop algorithms over graphs and then use data structures that are efficient for those algorithms
• Many important problems can be solved by:
  • Formulating them in terms of graphs
  • Applying a standard graph algorithm
• To make the formulation easy and standard, we have a lot of standard terminology about graphs

• In undirected graphs, edges have no specific direction
  • Edges are always two-way
• Only one of these edges needs to be in the set \(E\) (the other is implicit)
• Degree of a vertex: number of edges containing that vertex
  • Put another way: the number of adjacent vertices
• A path is a sequence of vertices connected by edges, with no repeated edges
  • Two vertices are connected if there is a path between them
  • A graph is connected when there is a path between every two vertices
• A cycle is a path (with at least 1 edge) who first and last vertices are the same
  • i.e., a loop
  • A graph can by cyclic (is has cycles) or acyclic (it doesn't have cycles)

• In directed graphs (sometimes called digraphs), edges have a direction
• Thus, just because the edge \((u,v)\) is in the graph doesn't mean that \((v,u)\) is in the graph.
• Call \(u\) the source and \(v\) the destination
• In-degree of a vertex: number of in-bound edges (i.e., edges where the vertex is the destination)
• Out-degree of a vertex: number of out-bound edges (i.e., edges where the vertex is the source)
• Directed graphs can have paths and cycles as well
  • Vertex \(w\) is reachable from vertex \(v\) if there is a directed path from \(v\) to \(w\)

• A node can have a degree / in-degree / out-degree of zero
• A graph does not have to be connected
  • Even if every node has non-zero degree
• For a graph \(G = (V,E)\):
  • \(|V|\) is the number of vertices
  • \(|E|\) is the number of edges
  • Minimum edges? \(0\)
  • Maximum for undirected? \(\frac{|V||V+1|}{2}\), which is \(O(|V|^2)\)
  • Maximum for directed? \(|V|^2 - |V|\), which is \(O(|V|^2)\)
    • (unless vertices can have self-edges, in which case it is \(|V|^2\))
• If \((u,v)\) is in \(E\)
  • Then \(v\) is a neighbor of \(u\), i.e., \(v\) is adjacent to \(u\)
  • Order matters for directed edges
  • \(u\) is not adjacent to \(v\) unless \((v,u)\) is in \(E\)