Graph Theory -Chapter 14

A circuit in a graph is

a path that begins and ends with the same vertex.

Brute Force Algorithm

solves a problem through exhaustion: it goes through all possible choices until a solution is found. The time complexity of a brute force algorithm is often proportional to the input size. Brute force algorithms are simple and consistent, but very slow.

Nearest Neighbor Algorithm

starts at a designated vertex and at each step it visits the nearest neighbor (among the vertices not yet visited) until the tour is completed.

Brute Force Algorithm

straightforward method of solving a problem that relies on sheer computing power and trying every possibility rather than advanced techniques to improve efficiency

Hamilton Circuit/Path

A Hamilton Path is a path that goes through every Vertex of a graph exactly once. A Hamilton Circuit is a Hamilton Path that begins and ends at the same vertex. *Unlike Euler Paths and Circuits, there is no trick to tell if a graph has a Hamilton Path or Circuit.

Hamilton Circuit

A Hamilton circuit in a graph is a circuit that visits each ver-tex exactly once (returning to the starting vertex to complete the circuit). Any complete graph with three or more vertices has a Hamilton circuit. In this text, Hamilton circuits that differ only in their starting points are considered to be the same circuit.

cycle

A circuit that that doesn't repeat vertices (dots) is called a

Coloring and Chromatic Number

A coloring for a graph is an assignment of a color to each vertex in such a way that vertices joined by an edge have different colors. The chromatic number of a graph is the least number of colors needed to achieve a coloring. Coloring a Graph Step 1 Choose a vertex with greatest degree, and color it. Use the same color to color as many vertices as you can without coloring two vertices the same color if they are joined by an edge. Step 2 Choose a new color, and repeat what you did in Step 1 for vertices not already colored. Step 3 Repeat Step 1 until all vertices are colored

Connected Graph (Alternative Definition)

A connected graph is one in which there is at least one path between each pair of vertices. Tree A graph is a tree if the graph is connected and contains no circuits.

Cut Edge

A cut edge in a graph is an edge whose removal disconnects a component of the graph

Connected and Disconnected Graphs

A graph is connected if we can move from each vertex of the graph to every other vertex of the graph along edges of the graph. If not, the graph is disconnected. The connected pieces of a graph are the components of the graph.

Spanning Tree and Minimum Spanning Tree

A spanning tree for a graph is a subgraph that includes every vertex of the original graph and is a tree. A spanning tree that has minimum total weight is a minimum spanning tree for the graph

Walks, Paths, and Circuits

A walk in a graph is a sequence of vertices, each linked to the next vertex by a specified edge of the graph. A path is a walk that uses no edge more than once. A circuit is a path that begins and ends at the same vertex. Complete Graph A complete graph is a graph in which there is exactly one edge going from each vertex to each other vertex

Euler Path and Euler Circuit

An Euler path in a graph is a path that uses every edge of the graph exactly once. An Euler circuit in a graph is a circuit that uses every edge of the graph exactly once. Euler's Theorem Suppose we have a connected graph. 1. If the graph has an Euler circuit, then each vertex of the graph has even degree. 2. If each vertex of the graph has even degree, then the graph has an Euler circuit

Euler Theorems for Graph Theory

An Eulerian trail exists in a connected graph if there are no odd vertices (dots) or two odd vertices.

Spanning Trees

Any tree that covers all vertices

Kruskal's Algorithm for Finding a Minimum Spanning Tree for Any Connected, Weighted Graph Choose edges for the spanning tree as follows. Step 1 First edge: Choose any edge with minimum weight. Step 2 Next edge: Choose any edge with minimum weight from those not yet selected. (At this stage, the sub-graph may look disconnected.) Step 3 Continue to choose edges of minimum weight from those not yet selected, except do not select any edge that creates a circuit in the subgraph. Step 4 Repeat Step 3 until the subgraph connects all vertices of the original graph

Choose edges for the spanning tree as follows. Step 1 First edge: Choose any edge with minimum weight. Step 2 Next edge: Choose any edge with minimum weight from those not yet selected. (At this stage, the sub-graph may look disconnected.) Step 3 Continue to choose edges of minimum weight from those not yet selected, except do not select any edge that creates a circuit in the subgraph. Step 4 Repeat Step 3 until the subgraph connects all vertices of the original graph

Fleury's Algorithm

Fleury's algorithm starts at an odd vertex if the graph has none. starts at an odd vertex if the graph has none.

Number of Vertices and Edges in a Tree

If a graph is a tree, then the number of edges in the graph is one less than the number of vertices. A tree with n vertices has n−1 edge

Unique Path Property of Trees

In a tree, there is always exactly one path from each vertex in the graph to any other vertex in the graph.

Minimum Hamilton Circuit

In a weighted graph, a minimum Hamilton circuit is a Hamilton circuit with the least possible total weight. Nearest Neighbor Algorithm Step 1 Choose a starting point for the circuit. Call this vertex A. Step 2 Check all the edges joined to A, and choose one that has the least weight. Proceed along this edge to the next vertex. Step 3 At each vertex you reach, check the edges from there to vertices not yet visited. Choose one with the least weight. Proceed along this edge to the next vertex. Step 4 Repeat Step 3 until you have visited all the vertices. Step 5 Return to the starting vertex.

How to complete a Hamilton circuits in a complete graph with 25 vertices.

Recall the way to find out how many Hamilton circuits this complete graph has. The complete graph above has four vertices, so the number of Hamilton circuits is: (N - 1)! = (4 - 1)!

Find a Euler Circuit on this graph using Fleury's algorithm

Start at any vertex if finding an Euler circuit. If finding an Euler path, start at one of the two vertices with odd degree. Choose any edge leaving your current vertex, provided deleting that edge will not separate the graph into two disconnected sets of edges. Add that edge to your circuit, and delete it from the graph. Continue until you're done.

Fleury's Algorithm

Step 1 Start at any vertex. Go along any edge from this vertex to another vertex. Remove this edge from the graph. Step 2 You are now on a vertex of the revised graph. Choose any edge from this vertex, subject to only one condition: Do not use a cut edge (of the revised graph) unless you have no other option. Go along your chosen edge. Remove this edge from the graph. Step 3 Repeat Step 2 until you have used all the edges and returned to the vertex at which you started.

How to determine the sum of the degrees of the vertices.

Sum of degree of all the vertices is twice the number of edges (lines) contained in it.

How to determine how many vertices (dots) are in a graph.

The maximum number of edges possible in a single graph with 'n' vertices is nC2 where nC2 = n(n - 1)/2.

Determine the number of edges (Line) in a tree with 50 vertices (dots).

The maximum number of edges possible in a single graph with 'n' vertices is nC2 where nC2 = n(n - 1)/2. The number of simple graphs possible with 'n' vertices = 2nc2 = 2n(n-1)/2.

How to determine how many edges the graph has.

The maximum number of edges possible in a single graph with 'n' vertices is nC2 where nC2 = n(n - 1)/2. The number of simple graphs possible with 'n' vertices = 2nc2 = 2n(n-1)/2.

Isomorphic Graphs

Two graphs are isomorphic if there is a one-to-one matching between vertices of the two graphs, with the property that whenever there is an edge between two vertices of either one of the graphs, there is an edge between the corresponding vertices of the other graph.

What is isomorphic in graph theory?

Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic.

A graph is a

collection of vertices (at least one) and edges. Each edge goes from a vertex to a vertex.

A graph is said to be connected

if any two of its vertices (dots) are joined by a path.

An Euler circuit in a graph

is a circuit that uses every edge of the graph exactly once.

A Hamilton circuit in a graph

is a circuit that visits each vertex exactly once (returning to the starting vertex to complete the circuit)

A disconnected graph

is a graph with multiple disconnected vertices (dots) and edges (lines).

Circuits

is a path that begins and ends at the same vertex (Dot).

An Euler path

is a path that uses every edge in a graph with no repeats. Being a path, it does not have to return to the starting vertex.

A path in a graph

is a sequence of vertices (Dots) connected by edges (Lines). A simple path is one with no repeated vertices.

Nonisomorphic Trees

is a special case of graph isomorphism. The word isomorphism means the same shape. So, intuitively, we say that two trees are isomorphic if they have the same structure.

Kruskal's Algorithm

is used to discover the shortest path between two points in a connected weighted graph.

Sum-of-the-Degrees Theorem In any graph,

the sum of the degrees of the vertices equals twice the number of edges

Two graphs are isomorphic if

there is a one-to-one matching between vertices, with the property that whenever there is an edge between two vertices of either one of the graphs, there is an edge between the corresponding vertices of the other graph.

What does leave your answer as factorial mean in math?

to multiply all whole numbers from our chosen number down to 1. Examples: 4! = 4 × 3 × 2 × 1 = 24.