Graph theory
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This document introduces graph theory and describes an algorithm for finding an Eulerian path in a graph. It defines what a graph is and some key graph concepts like nodes, edges, and neighbors. It then provides an informal "urban definition" and a formal definition of the Eulerian path problem. Finally, it outlines the steps of the algorithm to find an Eulerian path, which involves checking if the graph is valid, finding a starting node, using a stack to traverse edges until all are visited.
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Graph theory
- 1. Introduction to Graph Theory Eulerian path Algorithm
- 2. Graph theory A graph is used in mathematics and computer science to represent pairwise relations between objects. It is represented as a collection of nodes and edges. The edges connect the nodes. Each node or/and edge can have certain values or names assigned to it. Two nodes are called neighbors if they are connected with an edge. Two edges are connected if they have a common node. simple graph:
- 3. Example of a graph: In this example the nodes are the cities in Macedonia and the edges are the roads between a pair of cities.
- 4. Definition of the Eulerian path problem Urban definition: Given a drawing on a piece of paper, try to cross all the lines with a pen such that the pen doesnt leave the paper and you mustnt cross a line more than once. Professional definition: Given a graph G, find a sequence of connected edges S such that every edge of G is found in S. Try solving:
- 5. The Algorithm Step 1: we first have to input the graph, we will call the graph G. Step 2: next we have to check if G is valid. If G is not valid, no solution exists and stop the algorithm. How to check if G is valid: First we have to determine the number of neighboring nodes to every node of G, we will keep this numbers in the sequence X. Than we count how many odd numbers are there are in X, and we will call this number N. If N = 0 or N = 2 the graph is valid, else the graph is not valid. Step 3: next we have to find a starting node, we will name this node W. if N = 0 than set W to a random node If N = 2 than set W to a node with an odd number of neighbors Step 4: define a stack S and push W onto S, also define a sequence of nodes P, P will contain the eulerian path solution to G. Step 5: while S is nonempty repeat the following steps: if W has at least one neighbor Set O to the nearest node to W Disconnect O from W Set W to O Go to step 5 Else add W to P, set W to the top of S, Pop the first element from S Step 5: print P and stop the algorithm. e.g. graph