Partial ordering: A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive.
Partially ordered set (poset): A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S,R). Members of S are called elements of the poset.
Less/greater than or equal (\leq/\geq), inclusion relation (\subseteq), divisibility relation (|) are all partial orderings.
Less/greater than (</>) are antisymmetric and transitive, but not reflexive, so they are not partial orderings.
Comparable: The elements a and b of a poset (S,\preceq) are called comparable if either a\preceq{b} or b\preceq{a}. When a and b are elements of S such that neither a\preceq{b} nor b\preceq{a}, a and b are called incomparable.
Totally/linearly ordered set: If (S,\preceq) is a poset and every two elements of S are comparable, S is called a totally or linearly ordered set, and \preceq is called a total or linear order. A totally ordered set is also called a chain.
Well-ordered set: (S,\preceq) is a well-ordered set if it is a poset such that \preceq is a total ordering and every nonempty subset of S has a least element.
The principle of well-ordered induction: Suppose that S is a well-ordered set. Then P(x) is true for all x\in{S}, if (inductive step:) For every y\in{S}, if
P(x) is true for all x\in{S} with x\prec{y}, then P(y) is true.
Lexicographic ordering: The lexical ordering \prec on A_1\times{A}_2 is defined by specifying that one pair is less than a second pair if the first entry of the first pair is less than (in A_1) the first entry of the second pair, or if the first entries are equal, but the second entry of this pair is less than (in A_2) the second entry of the second pair.
Hasse diagram: Start with the directed graph for this relation. First, Remove these loops because of reflexivity. Next, remove all edges that must be in the partial ordering because of transitivity. Finally, arrange each edge so that its initial vertex is below its terminal vertex and remove all the arrows on edges.
Covers: An element y\in{S} covers an element x\in{S} if x\prec{y} and there is no element z\in{S} such that x\prec{z}\prec{y}.
Covering relation: The set of pairs (x,y) such that y covers x is called the covering relation of (S,\preceq).
Maximal element: An element of a poset is called maximal if it is not less than any element of the poset. The top element of a Hasse diagram.
Minimal element: An element of a poset is called minimal if it is not greater than any element of the poset. The bottom element of a Hasse diagram.
Greatest element: An element in a poset that is greater than every other element.
Least element: An element in a poset that is less than every other element.
Upper bound: Element greater than or equal to all the elements in a
subset A of S.
Lower bound: Element less than or equal to all the elements in a subset A of S.
Least upper bound: Upper bound that is less than every other upper bound of a subset A of S.
Greatest lower bound: Lower bound that is greater than every other lower bound of a subset A of S.
Lattice: A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice.
(P(S),\subseteq/\supseteq) is a lattice, with LUB and GLB being A\cup{B}
and A\cap{B}.
Compatible: A total ordering \preceq said to be compatible with the partial ordering R if a\preceq{b} whenever aRb.
Topological sorting: Constructing a compatible total ordering from a partial ordering.
Lemma 1: Every finite nonempty poset (S,\preceq) has at least one minimal element.
Algorithm for topological sorting: Pick the least element and remove it from the poset. Can also be done with a Hasse diagram.
Graphs
(Undirected) graph: A graph G=(V,E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to
connect its endpoints.
Simple graph: A graph in which each edge connects two different vertices and where no two edges connect the same pair of vertices is called a simple graph.
Infinite graph: A graph with an infinite vertex set or an infinite number of edges is called an infinite graph.
Finite graph: a graph with a finite vertex set and a finite edge set is called a finite graph.
Multigraph: Graphs that may have multiple edges connecting the same vertices are called multigraphs.
Loop: Edges that connect a vertex to itself.
Pseudographs: Graphs that may include loops, and possibly multiple edges connecting the same pair of vertices or a vertex to itself.
Directed graph (digraph): A directed graph (or digraph) (V,E) consists of a nonempty set of vertices V and a set of directed edges (or arcs) E. Each directed edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair (u,v) is said to start at u and end at v.
Simple directed graph: A directed graph with no loops and no multiple directed edges that start and end at the same vertices.
Directed multigraphs: Directed graphs that may have multiple directed edges from a vertex to a second (possibly the same) vertex.
Multiplicity: When there are m directed edges, each associated to an ordered pair of vertices (u,v), we say that (u,v) is an edge of multiplicity m.
Mixed graph: A graph with both directed and undirected edges.
Graph Terminology and Special Types of Graphs
Adjacent (Neighbor): Two vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if u and v are endpoints of an edge e of G. Such an edge e is called incident with the vertices u and v and e is said to connect u and v.
Neighborhood: The set of all neighbors of a vertex v of G=(V,E), denoted by N(v), is called the neighborhood of v. If A is a subset of V , we denote by N(A) the set of all vertices in G that are adjacent to at least one vertex in A. So, N(A)=\bigcup_{v\in{A}}N(v).
Degree: The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex. The degree of the vertex v is denoted by deg(v).
Theorem 1, The handshaking theorem: Let G=(V,E) be an undirected graph with m edges. Then 2m=\sum_{v\in{V}}deg(v). (Note that this applies even if multiple edges and loops are present.)
Theorem 2: An undirected graph has an even number of vertices of odd degree.
Adjacent to/from, initial/terminal vertex: When (u,v) is an edge of the graph G with directed edges, u is said to be adjacent to v and v is said to be adjacent from u. The vertex u is called the initial vertex of (u,v), and v is called the terminal or end vertex of (u,v). The initial vertex and
terminal vertex of a loop are the same.
In/out degree: In a graph with directed edges the in-degree of a vertex v, denoted by deg^−(v), is the number of edges with v as their terminal vertex. The out-degree of v, denoted by deg^+(v), is the number of edges with v as their initial vertex. (Note that a loop at a vertex
contributes 1 to both the in-degree and the out-degree of this vertex.) Theorem 3: Let G=(V,E) be a graph with directed edges. Then
\sum_{v\in{V}}deg^−(v)=\sum_{v\in{V}}deg^+(v)=|E|.
Underlying undirected graph: The undirected graph that results from ignoring directions of edges is called the underlying undirected graph.
Complete graph: A complete graph on n vertices, denoted by K_n, is a simple graph that contains exactly one edge between each pair of distinct vertices.
Noncomplete graph: A simple graph for which there is at least one pair of distinct vertex not connected by an edge.
Cycle: A cycle C_n, n\geq3, consists of n vertices v_1,v_2,...,v_n and edges {v_1,v_2},{v_2,v_3},...,{v_{n−1},v_n},{v_n,v_1}.
Wheel: We obtain a wheel W_n when we add an additional vertex to a cycle C_n, for n\geq3, and connect this new vertex to each of the n vertices in C_n, by new edges.
n-Cube: An n-dimensional hypercube, or n-cube, denoted by Q_n, is a graph that has vertices representing the 2^n bit strings of length n.
Bipartite and bipartition: A simple graph G is called bipartite if its vertex set V can be partitioned into two disjoint sets V_1 and V_2 such that every edge in the graph connects a vertex in V_1 and a vertex in V_2 (so
that no edge in G connects either two vertices in V_1 or two vertices in V_2). When this condition holds, we call the pair (V_1,V_2) a bipartition of the vertex set V of G.
Theorem 4: A simple graph is bipartite if and only if it is possible to assign one of two different colors to each vertex of the graph so that no two adjacent vertices are assigned the same color.
Complete Bipartite Graph: A complete bipartite graph K_{m,n} is a graph that has its vertex set partitioned into two subsets of m and n vertices, respectively with an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset.
Bipartite graphs can be used to model many types of applications that involve matching the elements of one set to elements of another.
Regular graph: A simple graph is called regular if every vertex of this graph has the same degree. A regular graph is called n-regular if every vertex in this graph has degree n.
Subgraph: A subgraph of a graph G=(V,E) is a graph H=(W,F), where W\subseteq{V} and F\subseteq{E}. A subgraph H of G is a proper subgraph of G if H=G.
Subgraph induced by vertex set: Let G=(V,E) be a simple graph. The subgraph induced by a subset W of the vertex set V is the graph (W,F), where the edge set F contains an edge in E if and only if both endpoints of this edge are in W.
Spanning subgraph: H is a spanning subgraph of G if W=V, F\subseteq{E}.
Union of graph: The union of two simple graphs G_1=(V_1,E_1) and G_2=
(V_2,E_2) is the simple graph with vertex set V_1\cup{V}_2 and edge set
E_1\cup{E}_2. The union of G_1 and G_2 is denoted by G_1\cup{G}_2.