\magnification 2050
\parindent 0pt
\parskip 10pt
\hsize 7.2truein
\vsize 9.7 truein
\hoffset -0.45truein
\def\u{\vskip -7pt}
\def\v{\vskip -5pt}


HW 2.1:  2, 4, 9, 11 and 2.2:  1, 2, 3-7, 12, 14, 17, 


Defn:  A graph is a pair of sets $(V, E)$ where the elements of $V$ are called vertices and the elements of $E$ are unordered pairs of elements of $V$.  The elements of $E$ are called edges.

Defn:  A directed graph or digraph is a pair of sets $(V, E)$ where the elements of $V$ are called vertices and the elements of $E$ are ordered pairs of elements of $V$.  The elements of $E$ are called arcs.

Defn:  A digraph is said to have multiple arcs if there exists more than one arc from $u$ to $v$.  

Hence a digraph that contains two arcs from $u$ to $v$ has multiple arcs, but a digraph that
contains exactly one arc from $u$ to $v$ and one arc from $v$ to $u$ does NOT have multiple arcs.

Defn: A graph is said to have multiple edges if there exists more than one edge between $u$ and $v$.  
 
Note the above definitions mean that we are assuming (unless otherwise stated) that graphs do not contain multiple edges and digraphs do not contain multiple arcs.

In chapters 2 and 3, we will also assume (unless otherwise stated) that graphs and digraphs have a finite number of vertices and do not contain loops.


2.2:  Connectedness


table 2.1


$v$ reachable from $u$ if there exists a path from $u$ to $v$.

Thm 2.1:  If $v$ is reachable from $u$, then there is a simple path from $u$ to $v$.


Proof 1 (ugly version):  If $v$ is reachable from $u$, there exists a path $u = u_1, u_2, ..., u_{n-1}, u_n = v$

If $u_1 = u_2$ let $p_1 =$ the path $u_2, ..., u_n$ 
\u
If $u_1 \not= u_2$ let $p_1 =$ the path $u_1, u_2, ..., u_n$ 


Suppose the path $p_{k-1} = u_{m_1}, ..., u_{m_j}, u_{k}, u_{k+1}, ..., u_n$
is defined such that 

\u
~~~~~~~~~
1.) $m_i < m_r$ if $i < r$.

\u
~~~~~~~~~
2.)  $m_i < k $ for $i = 1, ..., j$.

\u
~~~~~~~~~
3.)  $u_{k} \not= u_{m_i}$ for $i = 1, ..., j$.



If $u_k = u_{m_i}$  for some $i = 1, ..., j$

\centerline{let $p_k =$ the path $u_{m_1}, ..., u_{m_i}, u_{k+1}, ..., u_n$}
\u

If $u_k \not= u_{m_i}$  for some $i = 1, ..., j$

\centerline{let $p_k =$ the path $u_{m_1}, ..., u_{m_j}, u_k, u_{k+1}, ..., u_n$}


The path $p_{n-1}$ is a simple path from $u$ to $v$.





$v$ joined to $u$ if there exists a semipath between $u$ and $v$.


2.2.2 Distance

The length of a path is the number of arcs in the path

If $v$ is reachable from $u$, $d(u, v) = $ the distance between $u$ and $v$ = the length of a shortest path from $u$ to $v$ 


Note

0.) $d(u, v)$ is not defined if $v$ is NOT reachable from $u$.

1.)  $d(u, u) = 0$.

2.)  $d(u, v) \not= d(v, u)$.

3.)  If $w$ is reachable from $u$ and $v$ is reachable from $w$, then $v$ is reachable from $u$ and
$d(u, v) \leq d(u, w)  + d(w, v)$.


2.2.4 Connectedness Categories

A digraph is strongly connected or strong if for every pair of vertices $u$ and $v$, $u$ is reachable from $v$ AND $v$ is reachable from $u$.

A digraph is unilaterally connected or unilateral if for every pair of vertices $u$ and $v$, $u$ is reachable from $v$ OR $v$ is reachable from $u$.


A digraph is weakly connected or weak if every pair of vertices $u$ and $v$ is joined.


A digraph is disconnected if it is not weakly connected.

A digraph is connected of
\v
~~~~~~~~
	degree 0 if it is disconnected
\v
~~~~~~~~
	degree 1 if it is weak, but not unilateral
\v
~~~~~~~~
	degree 2 if it is unilateral, but no strong
\v
~~~~~~~~	degree 3 if it is strong





2.2.5  Criteria for Connectedness

Thm 2.3:  A digraph is strongly connected if and only if it has a complete closed path.

Lemma:  In any set of vertices of a unilateral digraph D, there is a vertex which can reach all others in the set.

Thm 2.4:  A digraph is unilaterally connected if and only if it has a complete path.


Thm 2.5:  A digraph is weakly connected if and only if it has a complete semipath.


2.2.6:  A graph is connected if between any pair of vertices, there is a chain. Thus a graph is 
connected if and only if it has a complete chain.

\end



A path is a sequence of $u_1, (u_1, u_2), u_2, (u_2, u_3), u_3,  ..., u_{n-1}, (u_{n-1}, u_n), u_n$ where $u_i \in V$ and $(u_i, u_{i+1}) \in E$.

Or equivalently a path is a sequence of vertices $u_1, u_2, u_3, ..., u_n$ where $(u_i, u_{i+1}) \in E$.

The path $u_1, u_2, u_3, ..., u_n$ is closed if $u_1 = u_n$.





Pop quizzes (if you read this, make sure you read the last sentence).

We probably won't have time for more than one pop quiz.

But giving exactly one pop quiz is impossible:

Proof:

Suppose that the pop quiz is not given until the last day of class.  The pop quiz is not a pop quiz