Lesson 22: Basic Graph Concepts

Transcription

1 Lesson 22: asic Graph oncepts msc 175 iscrete Mathematics 1. Introduction graph is a mathematical object that is used to model different relations between objects and processes: Linked list Flowchart of a program Structure chart of a program Finite state automata ity map Electric circuits ourse curriculum efinition: graph is a collection (nonempty set) of vertices and a set (possibly empty) of edges Vertices (also called nodes): can have names and properties Edges (also called links): connect two vertices, can be labeled, can be directed Each directed edge has a start vertex and an end vertex Loop: an edge that connects a vertex to itself Parallel edges: two or more distinct edges with same start and end. Example: Vertices: Edges:,,,,,, 1

2 Same graph given in another way: 2. asic oncepts The variety of graphs is due to the variety of way we can connect the vertices in a graph Loops and parallel edges: e4 and e5 are parallel edges, e6 is a loop e1 e3 e6 e2 e4 e5 Simple graph: a graph without loops and without parallel edges djacent edges: incident on one and the same vertex, e.g. e2 and e3, e2 and e6, etc: e1 e2 e3 e6 e4 e5 2

3 Isolated vertex: no edges are incident on that vertex djacent vertices: connected by an edge path from vertex x to vertex y : a list of vertices in which successive vertices are connected by edges Graph1: Some paths in Graph1 are: Simple path: No vertex is repeated. Length of a path: the number of edges connecting the vertices in the path Examples: is a simple path, while is not a simple path ycle: Simple path except that the first vertex is equal to the last. Examples:,,,,, 3

4 3. Types of Graphs Simple graphs: Graphs without loops and without parallel edges onnected graph: There is a path between each two vertices Graph1 is a connected graph. Examples of disconnected graphs: Graph2 Graph3 Vertices:,,, Vertices:,,, Edges:, Edges:, omplete graphs: Graphs with all edges present each vertex is connected to all other vertices. Example: E ense graphs: relatively few of the possible edges are missing Sparse graphs: relatively few of the possible edges are present 4

5 irected graphs: The edges are oriented, they have a beginning and an end. Sometimes the edges of a directed graph are called arcs. Examples: Graph4: Graph5: Graph4 and Graph5 are different graphs Weighted graphs weights are assigned to each edge (e.g. road map with distances) Networks: weighted graphs or directed weighted graphs Examples: E E 9 3 5

6 4. Subgraphs Subgraph K of a graph G: a graph with the following properties: Each vertex in K is a vertex in G. Each edge in K is an edge in G with the same end-points. Example: Graph: Subgraphs: 6

7 5. Spanning tree of a graph Tree: graph with no cycles Note: in a tree, when we choose a root we impose an orientation. We may choose any node to be the root, E.G: a b c d e If we choose vertex a to be the root, then the tree will look like: a c e b If we choose c to be the root, then the tree will be: d c a e b d 7

8 Note also, that the graph does not specify the order of the children of a given node. spanning tree of a graph: a subgraph that contains all the vertices, and no cycles If we add any edge to the spanning tree, it forms a cycle, and the tree becomes a graph. Example: Spanning trees for Graph1: tree with N vertices has N-1 edges. graph with less than N-1 edges is not connected. 8

9 6. Graph representation in programming There are two commonly used methods for graph representation: djacency matrix representation djacency-lists representation The choice depends on the type of the graph whether it is dense or not. 1. djacency matrix representation. This method is used when the graph is dense relatively few of all possible links are missing. The representations consists in building a matrix with number of rows equal to the number of columns, equal to the number of graph nodes. For the adjacency matrix the following is true: (i,j) = 1 if there node i and node j are connected with an edge. 0 otherwise. The diagonal elements (i,i) can be set either to 1 or to 0, whatever is more appropriate for the particular application. Example: Graph G: vertices,,,,e Edges: (,), (,), (,),(,E),(,), (,E) is represented by the following adjacency matrix: E E When implementing this representation it is convenient to assign integers to the vertices. If the number of nodes is V, the matrix requires V 2 bits of storage and V 2 initialize it. steps to 2. djacency-lists representation. This representation is used when few of all possible edges are present i.e. the graph is not dense. For each node a list is created that holds all nodes, connected to it. 9

10 For the graph from the above example the adjacency list representation will be the following: Graph G: vertices,,,,e Edges: (,), (,), (,),(,E),(,), (,E) :, :,, E :,, E :, E:, The adjacency list representation is better for sparse graphs because the space required is O(V+E), as contrasted to O(V 2 ) for the matrix representation. 7. List of terms Graph: collection (nonempty set) V of vertices and E of edges. E may be empty. Notation G(V, E). Subgraph: Let G (V, E) is a graph. Gs (Vs, Es) is a subgraph of G iff Vs is a subset of V, Es is a subset of E, containing those edges in E that connect the vertices of Vs in the graph G Loop: n edge that connects a vertex with itself Parallel edges: distinct edges with same end-points n edge is incident on each of its end-points djacent edges: incident on one and the same vertex djacent vertices: connected by an edge Isolated vertex: no edges are incident on that vertex Path from vertex x to vertex y : a list of vertices starting with x and ending in y in which successive vertices are connected by edges Simple path: No vertex is repeated. Length of a path: the number of edges connecting the vertices in the path ycle: path with no repeated vertices except for the first vertex and the last vertex, which are the same Tree unrooted: graph with no cycles spanning tree of a graph: subgraph that contains all the vertices, and no cycles 10

11 Types of Graphs Simple graphs: Graphs without loops and without parallel edges omplete graphs: Graphs with all edges present each vertex is connected to all other vertices. Weighted graphs weights are assigned to each edge (e.g. road map with distances) irected graphs: The edges are oriented, they have a beginning and an end. Sometimes the edges of a directed graph are called arcs. onnected graph: Undirected: irected: There is a path between every two vertices. There is a path between every two vertices in the underlying undirected graph isconnected graph: graph that is not connected graph with N vertices and less than N-1 edges is disconnected. Gs irected cyclic Graphs, directed graphs without cycles Networks: irected or undirected weighted graphs 11