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$G' = (V', E')$ is a subgraph of $G = (V, E)$ if $V'  \subset V$, $E' \subset E$, and $G'$ is a graph.

$G[V'] =(E', V')$  the subgraph of $G$ induced or spanned by $V'$ if $E' =\{ xy \in E ~|~ x, y \in V' \}$.


$G' = (V', E')$ is a spanning subgraph of $G = (V, E)$ if $V' = V$.

$G - W = G[V - W]$, $G - E' = (V, E - E')$, $G + xy = (V, E \cup \{xy\})$ where $x, y$ are nonadjacent vertices in $V$.


$|G| = $ order of $G = |V(G)|$ = number of vertices. 

$e(G) = $ size of $G = |E(G)|$ = number of edges. 

$G^n$ is a graph of order $n$, $G(n, m)$ is a graph of order $n$ and size $m$.


$E(U, V)$ = set of $U - V$ edges = set of all edges in $E(G)$ joining a vertex in $U$ to a vertex in $V$ where $U \cap V = \emptyset$.

The complement of $G = (V, E) = \overline{G} = (V, V^{(2)} - E)$

$K_n = $ complete graph on $n$ vertices.  $E_n = \overline{K_n}$ = empty graph with $n$ vertices.  $K_1 = E_1$ is trivial.


$\Gamma(x) =\Gamma_G(x) =  \{ y ~|~ xy \in E(G) \}$.

$d(x) = d_G(x) = deg(x) = $ degree of $x = |\Gamma(x) |$.


$\delta(G) = min \{ d(x) ~|~ x \in V(G) \}$.

$\Delta(G) = max \{ d(x) ~|~ x \in V(G) \}$.

$v$ is an isolated vertex if $d(v) = 0$.


$\Sigma_{x \in V} d(x) = 2e(G)$.

A walk in a graph, $W = v_0, e_1, v_1, e_2, ...., e_n, v_n$,  where $v_i \in V$ and $e_i = v_{i-1} v_i \in E$.

length of $W = n$.

trail =  walk with distinct edges.

 circuit = closed trail.

 path =  walk with distinct vertices ( = trail with distinct vertices).

 cycle = circuit with distinct vertices.


A set of vertices (edges) is independent if no two elements in the set are adjacent

A set of paths is independent if no two paths share an interior vertex.

$d(x, y) $ = length of shortest $x-y$ path.  If there is no $x-y$ path, then $d(x, y) = \infty$.

A graph is connected if given any pair of distinct vertices, $x, y$, there is an $x-y$ path.

A component of a graph =  a maximal connected subgraph.

A cutvertex = a vertex whose deletion increases the number of components.

A bridge = an edge  whose deletion increases the number of components.

A forest = an acyclic graph = a graph without any cycles.

A tree = a connected forest.

$G = (V, E)$ is bipartite if there exists vertex classes, $V_1, V_2$, such that $V_1 \cup V_2 = V$, $V_1 \cap V_2
 = \emptyset$, and $xy \in E$, $x \in V_i$ implies $y \not\in V_i$ (i.e., no edge joins two vertices in the same class).  

$K(n_1, ..., n_r) =$ complete r-partite graph.  $K_{p, q} = K(p, q)$, $K_r(t) = K(t, t, ..., t)$

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Thm 3:  Suppose that $C = (W, E')$ is the component of $G = (V, E)$ containing the vertex $x$.  Then
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$W = \{y \in V ~|~ G$ contains an $x-y$ path $\}$ $ = \{y \in V ~|~ G$ contains an $x-y$ trail $\}$ \hfil \break
$= \{y \in V ~|~ d(x,y) < \infty \}$ = equivalence class of x where we take the smallest equivalence relation on $V$ such that $u$ is equivalent to $v$ if $uv \in E$.


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HW  p. 28: 1

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