Graph Concepts Illustrated Using The Leda Library

Graph ConceptsIllustrated Using The Leda Library

Amanuel Lemma CS252 Algorithms

Vertices and Edges•Vertices or nodes store information and each edge

connects a pair of vertices.

•Drawn from: ../../handout/demo/graphwin/gw

Multiple Edges and Loops• Multiple edges(i.e parallel edges) and loops have the same beginning and end vertices

• Drawn from : ../../handout/demo/graphwin/gw

Undirected Graph• Def: set of vertices and a set of edges (each is a set of two vertices)


Directed Graph(digraph)• Def: A set of vertices and a set of edges(each is an ordered pair of vertices)


Simple Graph• Def : A graph with out loops and multiple edges

• From left to right: a simple undirected graph and a simple directed graph


Examples of Graphs and Multigraphs• Multigraph as opposed to a simple graph has multiple edges b/n any two nodes

• From left to right : a normal graph and a multigraph


Special Classes of Graphs : Complete and Bipartite• Complete graphs : each vertex has at least one edge going to every other vertex.

• Bipartite graphs : vertices can be divided into two classes with no edges with in class.

• From left to right : a complete graph on 5 vertices( K5) and a bipartite graph


Path in undirected graph• A sequence of vertices (v1,v2,…,vn) where there is an edge b/n vi and vi+1

• Drawn from : ../../handout/demo/xlman/graphwin

Path in a Directed graph•A sequence of vertices where there is an out-going edge b/n vi and vi+1


Hamilton Path in an Undirected Graph• A path in an undirected graph that spans or visits all the vertices

• Drawn from : ../../handout/demo/xlman/graphwin

Hamilton Path in a Directed Graph• A path in a directed graph that spans or visits all the vertices


Cycle in an Undirected Graph• A path in an undirected graph where the start and end vertex is the same (v0 = vn)


Cycle in a Directed Graph• A path in a directed graph where the start and end vertices are the same (v0 = vn)


Hamilton Cycle in an Undirected Graph• A Hamilton path in an undirected graph where the start and end vertices are the same.

• Drawn from ../../handout/demo/graphwin/gw

Hamilton cycle in a Directed Graph• A Hamilton path in a directed graph where the start and end vertices are the same


Cyclic and Acyclic Digraph• A digraph containing at least one cycle is a cyclic digraph.

• A digraph containing no cycles at all is an acyclic digraph.

• From left to right : an acyclic digraph and a cyclic digraph


**A Graph Which is not strongly Connected**

• There exsist a pair of vertices which have no directed path b/n them. Or

• A graph which can be decomposed in to two or more strongly connected components


More on strongly connected components• The program at “../../handout/demo/graph_alg/gw_scc” illustrates strongly connected components(scc) by coloring the nodes and giving the same number.

Tree• A connected graph with out cycles, loops and multi-edges.


Forest• A collection of trees(defined earlier). Or

• A graph with out cycles, loops and multiedges


Graph Concepts Illustrated Using The Leda Library

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