Introduction to Graph Theory

The textbook uses the following graph to introduce terms in graph theory:

e   ----------------------------              -------- 
 3 |                            |            |     e  |
   |      e            e        v            v      5 |
  ----     1    ----    2     ----          ----      |
 | v  | -----> | v  | -----> | v  |        | v  | ---- 
 |  1 |        |  2 |        |  3 |        |  4 |      
  ----          ----          ----          ----       
   |      e                     ^                      
   |       4                    |                      
    ----------------------------

Figure 1: A directed multigraph.
[Figure 2.2 from Software Synthesis from Dataflow Graphs.] A directed multigraph is an ordered pair (V,E) where V and E are finite sets called vertices and edges, respectively.

Edges
- edge: connects two vertices where each edge e is directed from a source vertex src(e) to a destination or sink vertex snk(e).
- output edge: e is the output edge of src(e)
- input edge: e is the input edge of snk(e)
Vertices
- predecessor: v₁ is a predecessor of v₂ if there is an edge e such that src(e) = v₁ and snk(e) = v₂.
- successor: v₁ is a successor of v₂ if v₂ is a predecessor of v₁
- adjacent: v₁ and v₂ are adjacent if v₁ is a successor or predecessor of v₂.
- chain: a sequence of vertices v₁, v₂, ..., v_k is a chain that joins v₁ and v_k if v_i+1 is adjacent to v_i for i = 1, 2, ..., k-1.
Graphs
- subgraph: a subgraph of a directed multigraph consists of a subset of vertices and all edges that connect the subset of vertices
- path: a path p in (V,E) is a non-empty sequence of edges p = e1, e2, e3, ... in E such that snk(e1) = src(e2), snk(e2) = src(e3) = ...
- associated graph: the edges in a path and the vertices associated with the edges in the path, but it is not necessarily a subgraph because some of the edges connecting the vertices may not be in the path.
- (directed) cycle: a path that is directed from some vertex to itself
- fundamental cycle: a cycle of which no proper subsequence is also a cycle
- ancestor: v₁ is an ancestor of v₂ if there is a path from v₁ to v₂, and the set of ancestors of vertex v is denoted by ancs(v)
- descendant: v₂ is a descendant of v₁ if there is a path from v₁ to v₂, and the set of descendants of vertex v is denoted by desc(v)
- independent: v₁ and v₂ are independent if v₁ is neither an ancestor or a descendant of v₂, and the set of vertices that are independent of v is denoted by independent(v)
- transitive: an edge is transitive if there is another path between the source and sink vertices of the edge.

Here are more definitions about graphs:

connected: a directed multigraph is connected if for any pair of distinct vertices v₁ and v₂, there is a chain that joins v₁ and v₂; for example, graph in Figure 1 is not connected, but the subgraph associated with {v₁, v₂, v₃} is connected.
partition: a partition of a set is a collection of disjoint subsets whose union is equal to the original set.
unique partition: every multigraph (V,E) has a unique partition V₁, V₂, ..., V_N such that
1. subgraph(V_i) is connected for every i, and
2. for each edge e in E, both src(e) and snk(e) are in the same partition V_j for some j
connected component: each vertex set V_i in a unique partition is a connected component; for example, the connected components in the graph in Figure 1 are {v₁, v₂, v₃} and {v₄}.
strongly connected: a directed multigraph of (V,E) is strongly connected if for each pair of distinct vertices v₁ and v₂, there is a path directed from v₁ to v₂ and from v₂ to v₁.
strongly connected component: a strongly connected component of (V,E) is a strongly connected subset V' of V such that no strongly connected subset of V properly contains V'.

The graph in Figure 2 below has three strongly connected components given by {v₁, v₂}, {v₃, v₄, v₅}, and {v₆}. The set {v₄, v₅} is strongly connected but is not a strongly connected component.

     ----              ----              ---- 
 -> | v  | ---------> | v  | ---------> | v  |
|   |  1 |            |  3 | <--        |  6 |
|    ----              ----     \        ---- 
|     |                 |        \         ^  
|     |                 |          ---     |  
|     |                 |             \    |  
|     v                 v              \   |  
|    ----              ----              ---- 
 -- | v  |            | v  | ---------> | v  |
    |  2 |            |  4 |            |  5 |
     ----              ----              ----
                        ^                  |  
                        |                  |  
                         ------------------

Figure 2: A directed multigraph.
[Figure 2.2 from Software Synthesis from Dataflow Graphs.] How many strongly connected components are there in the following graph? What are they?

e   ------------    ------------   e  
 4 |            |  |            |   3 
   v            |  v            |     
  ----          ----          ----    
 | v  | -----> | v  | -----> | v  |   
 |  1 |        |  2 |        |  3 |   
  ----          ----          ----    
   |            ^  |            ^     
e  |            |  |            |   e 
 1  ------------    ------------     2

Figure 3: Example of the concept of strongly connected component.

Updated 05/04/04.