Excursions in Modern Mathematics

Excursions in Modern Mathematics Chapter 5 Euler Circuits Sixth Edition The Circuit Comes to Town 1 2 Euler Circuits Outline/learning Objectives Euler Circuits Outline/learning Objectives (cont.) To identify and model Euler circuit and Euler path problems. To understand the meaning of basic graph terminology. To classify which graphs have Euler circuits or paths using Euler’s circuit theorems. 3 To implement Fleury’s algorithm to find an Euler circuit or path when it exists. To eulerize or semi-eulerize graphs when necessary. To recognize an optimal eulerization (semieulerization) of a graph. 4 Euler Circuits Euler Circuits What is a routing problem? Existence question Is an actual route possible? Optimization question Of all the possible routes, which one is the optimal route? 5.1 Euler Circuit Problems 5 6 1 Euler Circuits Euler Circuits We will answer both the existence and optimization questions for a special class of routing problems known as Euler circuit problems. The common thread is what we call the exhaustion requirement. The name of the game is to trace each drawing without lifting the pencil or retracing any of the lines. These kinds of tracings are called unicursal tracings. 7 8 Euler Circuits Euler Circuits 5.2 Graphs When we end in the same place we started, we call it a closed unicursal tracing; when we start and end in different places, we call it an open unicursal tracing. . 9 10 Euler Circuits Euler Circuits This graph has six vertices A, B, C, D, E, and F and eight edges. The edges can be described by giving the two vertices that are connected by the edge. Thus the edges are AB, AD, BB, BC, BE, CD, CD, and DE A graph is a picture consisting of: Vertices- dots Edges- lines The edges do not have to be straight lines. But they have to connect two vertices. Loop- an edge connecting a vertex back with itself 11 12 2 Euler Circuits 13 First, note that the point where edges BE and AD cross is not a vertex– it is just the crossing point of two edges. Second, that vertex F is not connected to any other vertex. Such a vertex is called an isolated vertex. Euler Circuits 14 Euler Circuits 15 This graph is considered a single graph, even though it consists of two separate, disconnected pieces. Such graphs are called disconnected graph, and the individual pieces are called the components of the graph.. Euler Circuits A Graph with No Edges? Yes, its possible. Without edges, every vertex of the graph is an isolated vertex. 16 Euler Circuits Euler Circuits Graphs A graph is a structure that defines pairwise relationships within a set to objects. The objects are the vertices, and the pairwise relationships are the edges: X is related to Y if and only if XY is an edge. 17 Third, note that this graph has a loop, namely the edge BB. Finally, note that it is permissible to have two edges connecting the same two vertices, as in the case with C and D. When a graph has more than one edge connecting the same pair of vertices, it is said to have multiple edges. 5.3 Graph Concepts and Terminology 18 3 Euler Circuits 19 Euler Circuits Adjacent vertices. Adjacent edges. Two vertices are said to be adjacent if there is an edge joining them. Vertices B and E are adjacent; C and D are not. Also because of the loop at E, we can say that Vertex E is adjacent to itself. Two edges are adjacent if they share a common vertex. AB and AD are adjacent; edges AB and DE are not. 20 Euler Circuits 21 Euler Circuits Degree of a vertex. Odd and even vertices. The degree of a vertex is the number of edges at that vertex. When there is a loop at the vertex, the loop contributes twice. The deg(A) = 3, deg(B) = 5, deg(C) = 3, deg(D) = 2, deg(E) = 4, etc. An odd vertex is a vertex of odd degree; an even vertex is a vertex of even degree. The graph has two even vertices (D and E) and six odd vertices (all the others). 22 Euler Circuits Euler Circuits Paths. 23 A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the next one. The key requirement in a path is that an edge can be part of a path only once. Paths (continued). 24 The number of edges in the path is called the length of the path. A, B, E, D. This is a path from vertex A to D, consisting of the edges AB, BE, and ED. The length of this path is 3. 4 Euler Circuits Euler Circuits Connected graphs. Circuits. A circuit has the same definition as a path, but has the additional requirement that the trip starts and ends at the same vertex. 25 26 Euler Circuits 27 A graph is connected, if given any two vertices, there is a path joining them. A graph that is not connected is said to be disconnected. A disconnected graph is made up of separate components. Euler Circuits Bridges. Euler paths. Sometimes in a connected graph there is an edge such that if we were to erase it, the graph would become disconnected—such an edge is called a bridge. BF, FG, and FH are bridges. An Euler path is a path that passes through every edge of a graph once and only once. The graph shown in (a) does not have an Euler path; the graph in (b) has several Euler paths. One of them is L,A,R,D,A,R,D,L,A. 28 Euler Circuits Euler circuits. 29 An Euler circuit is a circuit that passes through every edge of a graph. One of them is L,A,R,D,A,R,D,L,A,L. Note that if a graph has an Euler circuit it cannot have an Euler path, and vice versa. 5