Mathematics has played a vital role in the evolution of our society. The proposition of various mathematical theories have led us to travel to the moon, search out the secrets of DNA, transferring electricity over thousands of miles and giving rise to computers which has become the basic necessity of the modern world. Graph theory deals with the study of the mathematical structures that are used to model pairwise relations between objects. Graphs are a major field of study in discrete mathematics. The feature covers the route problems in graph theory with the help of relevant animations.

## Route Problems

### 1) Hamiltonian Path Problem

Image Courtesy: mathforum.org

Image Courtesy: combinatorica.com

This route problem determines whether a Hamiltonian path exists in a particular directed or undirected graph. A Hamiltonian path is the one in which the path visits each vertex only once in a directed or undirected graph. The problem can be solved by using n-Hamiltonian path computations where n is the number of vertices in the graph. The animations show the Hamiltonian path and Hamiltonian cycle.

### 2) Minimum Spanning Tree

Image Courtesy: combinatorica.com

It is an edge weighted graph in which one associates the weights with each edge. A minimum spanning tree of an edge-weighted graph is a spanning tree whose sum of the weights of its edges is no larger than the weight of any other spanning tree. A spanning tree connects all of the nodes in a graph and has no cycles. The animation is that of a minimum spanning tree.

### 3) Shortest Path Problem

Image Courtesy: Wikipedia.org

Image Courtesy: combinatorica.com

As quite indicative from the name, the shortest path problem concerns itself with finding the shortest path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized. It can be defined for directed, undirected or mixed graphs. Dijkstra’s graph search algorithm solves this shortest path problem by assigning non-negative edge path costs which leads to a shortest path tree as can be seen in the animation. The shortest path problem has various applications in web based mapping applications that provide the shortest route to a certain destination.

### 4) Steiner Tree Problem

Image Courtesy: Imgur.com

Set in a combinatorial condition, Steiner tree problem is a problem which requires the shortest interconnect for a given set of objects. Quite similar to the minimum spanning tree, the introduction of extra vertices or edges in the graph helps in reducing the length of the spanning tree and the resulting connection is termed as a Steiner tree. The animation shows the Steiner tree in which the red dots are input nodes and the blue dots are Steiner points. The Steiner tree problem has number of applications in network designs and circuit layouts.

### 5) Travelling Salesman Problem

Image Courtesy: cs.arizona.edu

This problem requires the shortest possible route such that a person visits each city exactly once and then returns to the origin point. It is one of the most studied problems in optimization even though it is computationally quite large. Methods are known such that the route to thousands of cities could be approximated within a small fraction of 1%. The travelling salesman problem has its application in the microchip manufacturing and DNA sequencing.