Lecture 10 INB371
Graphs are important in CS cause they can represent any relationship.
The key to solving problems is to think of them in terms of graphs, e.g. a GPS,
showing you a graph of point a to point b. There are lots of algorithms to solve
these sorts of problems, such as the shortest path algorithm by Djikstra and the
Minimum Spanning Tree algorithm.
Euler’s solution to the Konigsberg bridge problem brought about Graph Theory, an
entire new branch of mathematics. He did this by removing all physical properties of
the problem, leaving just the nodes and bridges.
When we have more than one pair between two nodes, it’s called a ‘parralell edge’.
If a node has an edge that connects to itself it’s called a ‘loop’. A sequence of edges
that connects two nodes is called a path.
Paths can be designated by the paths it crosses (wxyz), or BCDAC ( the vertices it
passes through). If a graph has parallel edges and you designate a path by the
vertices it passes through, it cannot determine which edge it goes down. A path is
called ‘closed’ if it starts and finishes at the same vertices. If it doesn’t repeat any
edge it’s called a ‘circuit’ or ‘cycle’. If no vertex is repeated the path is a ‘simple
Graphs can be categoriesd into undirected and directed. In undirected, the edges
don’t indicate a direction, and a directed graph does indicate direction. You can only
go that direction down the edge. A Subgraph consists of a subset of the graphs
vertices and a subset of the graphs edges. A weighted graph is a graphs whos
edges have a weight , e.g. it costs more to go down that edge. Vertices of a graph
are ‘adjacent’ if they’re connected by an edge. A graph is ‘connected’ if there’s a
path between each pair of distinct vertices (e.g. no vertices on their own with no
edges). A graph is complete if there’s an edge between each pair of vertices. A
degree of a vertex is the number of edges attached to it. A loop counts as two
An Euler Path is a path that uses every edge of a graph exactly once, starting and
ending at different vertices. An Euler circuit is a circuit that uses every edge of a
graph exactly once, ending on the same vertex it started on.
Criteria for Euler Paths
For every vertex on a particular graph other than the start/finish vertex, the path
enters the vertex the same number of times it leaves the vertex. Therefore, all
vertices other than the two endpoints of P must be even vertices, and the endpoints
of the path are odd vertices. Euler Circuits
For every vertex in the graph, each edge having v as an endpoint shows up exactly
once in the circuit.
The circuit enters vertex same number of times it leaves a vertex, so vertex has
degree 2s. therefore v must be an even vertex.
You can easily find out if a euler path or circuit exists by counting the amount of odd
There are four different representations of graphs:
There are two data structures to represent a graph,