Chapter 6

Bipartite Graph

a graph in which the vertices can be partitioned into two disjoint sets V₁ and V₂ such that each edge connects a vertex of V₁ to a vertex of V₂

Endpoints

points on the ends of edges

Connected

a graph is connected if there is a path between every pair of distinct vertices.

Acyclic

a graph that does not have a cycle.

Self Loop

A graph can also have a self-loop which is an edge between a vertex and itself. an edge with one endpoint

Undirected Graph

A graph in which the edges have no direction. In an undirected graph, the edges are unordered pairs of vertices, which is useful for modeling relationships that are symmetric. curly braces

Finite

A graph is finite if the vertex set is finite.

Euler Path

A path that contains all edges of a graphs.

Vertex

A single element of V is called a vertex and is usually represented pictorially by a dot with a label.

Subgraph

A subgraph S of a graph G is a graph whose set of vertices and set of edges are all subsets of G. Every graph is a subgraph of itself.

Walk

A walk from v₀ to vₙ in an undirected graph G is a sequence of alternating vertices and edges that starts and ends with a vertex: <v₀, {v₀, v₁}, v₁, {v₁, v₂}, v₂, ... , vₙ-₁, {vₙ-₁, vₙ}, vₙ> The vertices just before and after each edge are the two endpoints of that edge. Since the edges in a walk are completely determined by the vertices, a walk can also be denoted by the sequence of vertices: <v₀, v₁, v₂, ... , vₙ>

Regular Graph

All vertices have the same degree

Euler Circuit

An Euler circuit in an undirected graph is a circuit that contains every edge and every vertex. Note that a circuit, by definition, has no repeated edges, so an Euler circuit contains each edge exactly once. If an undirected graph G has an Euler circuit, then G is connected and every vertex in G has an even degree. Odd degree vertex implies no Euler circuit. If an undirected graph G is connected and every vertex in G has an even degree, then G has an Euler circuit. An undirected graph G has an Euler circuit if and only if G is connected and every vertex in G has even degree. Every vertex in the graph below has even degree. Therefore according to the sufficient conditions for an Euler circuit, the graph must have an Euler circuit.

Cycle

Cₙ is called a cycle on n vertices. The edges connect the vertices in a ring. Note that Cₙ is well defined only for n ≥ 2.

Edge

Each edge in E is a set of two vertices from V and is drawn as a line connecting the two vertices.

Procedure to Find a Circuit in a Graph

Find a vertex w, that is not an isolated vertex. Select any edge {w, x} incident to w. (Since w is not isolated, there is always at least one such edge.) Current trail T := <w, x> last := x While there is an edge {last, y} that has not been used in T: Add y to the end of T last := y

Adjacent

If there is an edge between two vertices, they are said to be adjacent.

Complete Graph

Kₙ is called the complete graph on n vertices. Kₙ has an edge between every pair of vertices. A complete graph on n vertices, denoted Kₙ, is a simple graph that contains exactly one edge between each pair of distinct vertices.

Number of Edges and Total Degree

Let G=(V, E) be an undirected graph. Then twice the number of edges is equal to the total degree: ∑ deg(v) 2 ⋅ |E| v∈V

Matrix Representation

The matrix representation for a graph with n vertices is an n by n matrix whose entries are all either 0 or 1, indicating whether or not each edge is present. If the matrix is labeled M, then M(sub i,j) denotes the entry in row i and column j.

Procedure to Find a Euler Circuit in a Graph

Use the procedure described above to find any circuit in G. Call the circuit C. The algorithm continues to iterate the following steps until all the edges in G are included in C: 1) Remove all edges in C from G. Remove any isolated vertices from G. Call the resulting graph G'. 2) Find a vertex w that is in G' and C. 3) Find a circuit in G' that begins and ends with w. Call the circuit C'. 4) Combine circuit C and C'. Suppose C starts and ends at vertex v. Create a new circuit that starts at v and follows the edges in C until w is reached. The new circuit then follows the edges in C' back to w and then follows the rest of the edges in C back to v. The new circuit is renamed C for the next iteration.

Neighbor

a vertex a is a neighbor of vertex b iff {a, b} is an edge

Incident

a vertex is incident to an edge if the vertex is one of the two vertices the edge connects.

D-Regular Graph

all the vertices have dregree d.

Clique

called a clique of size n or a n-clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete.

Adjacency List Representation

each vertex has a list of all its neighbors.

Simple Graph

if a graph does not have parallel edges or self-loops.

Degree

in a simple graph the degree of a vertex is the number of neighbors it has

Hamiltonian Cycle

in an undirected graph is a cycle that includes every vertex in the graph. Note that a cycle, by definition, has no repeated vertices or edges, except for the vertex which is at the beginning and end of the cycle. Therefore, every vertex in the graph appears exactly once in a Hamiltonian cycle, except for the vertex which is at the beginning and end of the cycle.

Hamiltonian Path

in an undirected graph is a path that includes every vertex in the graph. Note that a path, by definition, has no repeated vertices or edges, so every vertex appears exactly once in a Hamiltonian path. a simple path that contains all vertices of a graph.

Complete Bipartite Graph

is a bipartite graph in which each vertex in the first set is connected to each vertex in the second set by exactly one edge. If the first set has m elements the second set has n elements, then the complete bipartite graph is denoted Km,n

Cycle Walk

is a circuit of length at least 1 in which no vertex occurs more than once, except the first and last vertices which are the same. a path where the first and last vertices are the same.

Circuit

is a closed walk in which no edge occurs more than once. a path where the first and last vertices are the same.

Path

is a trail in which no vertex occurs more than once. sequence of n non repeated edges connecting two vertices is a walk of length n.

Open Walk

is a walk in which the first and last vertices are not the same.

Closed Walk

is a walk in which the first and last vertices are the same.

Euler Trail

is an open trail that includes each edge. Note that a trail, by definition, has no repeated edges, so an Euler trail contains each edge exactly once. In an open trail, the first and last vertices are not equal. An undirected graph G has an Euler trail if and only if G is connected and has exactly two vertices with odd degree.

Trail

is an open walk in which no edge occurs more than once.

Length of Walk

is l, the number of edges in the walk.

Total-Degree

is the sum of the degrees of all the vertices.

Parallel Edges

multiple edges between the same pair of vertices.

Undirected Graph Walk

v - w valid walks: <v,w> <w,v>

Directed Graph

x -> y valid walks: <x,y> x <-> y valid walks: <x,y> <y,x>