Lecture Notes CMSC 251 Lecture 19 Second Midterm Exam Tuesday April 7 1998 Second midterm exam today No lecture Lecture 20 Introduction to Graphs Thursday April 9 1998 Read Sections 5 4 5 5 Graph Algorithms For the next few weeks we will be discussing a number of various topics One involves algorithms on graphs Intuitively a graph is a collection of vertices or nodes connected by a collection of edges Graphs are very important discrete structures because they are a very flexible mathematical model for many application problems Basically any time you have a set of objects and there is some connection or relationship or interaction between pairs of objects a graph is a good way to model this Examples of graphs in application include communication and transportation networks VLSI and other sorts of logic circuits surface meshes used for shape description in computer aided design and geographic information systems precedence constraints in scheduling systems The list of application is almost too long to even consider enumerating it Most of the problems in computational graph theory that we will consider arise because they are of importance to one or more of these application areas Furthermore many of these problems form the basic building blocks from which more complex algorithms are then built Graphs and Digraphs A directed graph or digraph G V E consists of a finite set of vertices V also called nodes and E is a binary relation on V i e a set of ordered pairs from V called the edges For example the figure below left shows a directed graph Observe that self loops are allowed by this definition Some definitions of graphs disallow this Multiple edges are not permitted although the edges v w and w v are distinct This shows the graph G V E where V 1 2 3 and E 1 1 1 2 2 3 3 2 1 3 1 2 3 Digraph 1 3 2 4 Graph Figure 18 Digraph and graph example In an undirected graph or just graph G V E the edge set consists of unordered pairs of distinct vertices thus self loops are not allowed The figure above right shows the graph G V E where V 1 2 3 4 and the edge set is E 1 2 1 3 1 4 2 4 3 4 We say that vertex w is adjacent to vertex v if there is an edge from v to w In an undirected graph we say that an edge is incident on a vertex if the vertex is an endpoint of the edge In a directed graph we will often say that an edge either leaves or enters a vertex A digraph or undirected graph is said to be weighted if its edges are labelled with numeric weights The meaning of the weight is dependent on the application e g distance between vertices or flow capacity through the edge 60 Lecture Notes CMSC 251 Observe that directed graphs and undirected graphs are different but similar objects mathematically Certain notions such as path are defined for both but other notions such as connectivity are only defined for one In a digraph the number of edges coming out of a vertex is called the out degree of that vertex and the number of edges coming in is called the in degree In an undirected graph we just talk about the degree of a vertex as the number of edges which are incident on this vertex By the degree of a graph we usually mean the maximum degree of its vertices In a directed graph each edge contributes 1 to the in degree of a vertex and contributes one to the out degree of each vertex and thus we have Observation For a digraph G V E X X in deg v out deg v E v V v V E means the cardinality of the set E i e the number of edges In an undirected graph each edge contributes once to the outdegree of two different edges and thus we have Observation For an undirected graph G V E X deg v 2 E v V Lecture 21 More on Graphs Tuesday April 14 1998 Read Sections 5 4 5 5 Graphs Last time we introduced the notion of a graph undirected and a digraph directed We defined vertices edges and the notion of degrees of vertices Today we continue this discussion Recall that graphs and digraphs both consist of two objects a set of vertices and a set of edges For graphs edges are undirected and for graphs they are directed Paths and Cycles Let s concentrate on directed graphs for the moment A path in a directed graph is a sequence of vertices hv0 v1 vk i such that vi 1 vi is an edge for i 1 2 k The length of the path is the number of edges k We say that w is reachable from u if there is a path from u to w Note that every vertex is reachable from itself by a path that uses zero edges A path is simple if all vertices except possibly the first and last are distinct A cycle in a digraph is a path containing at least one edge and for which v0 vk A cycle is simple if in addition v1 vk are distinct Note A self loop counts as a simple cycle of length 1 In undirected graphs we define path and cycle exactly the same but for a simple cycle we add the requirement that the cycle visit at least three distinct vertices This is to rule out the degenerate cycle hu w ui which simply jumps back and forth along a single edge There are two interesting classes cycles A Hamiltonian cycle is a cycle that visits every vertex in a graph exactly once A Eulerian cycle is a cycle not necessarily simple that visits every edge of a graph exactly once By the way this is pronounced Oiler ian and not Yooler ian There are also path versions in which you need not return to the starting vertex 61