Graphs
From charlesreid1
Graphs are mathematical objects consisting of nodes and edges. The original inventor of graph theory was Leonhard Euler, who used it to solve the Seven Bridges of Königsberg problem.
Notes
Diestel - Graph Theory
Link: http://www.cs.unibo.it/babaoglu/courses/cas00-01/tutorials/GraphTheory.pdf
Chapter 1: Basics
Outline:
- Definitions (graph, degree, path, cycle, connectivity, tree, forest, k-partite, contraction, Euler tours)
Graph definitions
A graph G consist of a set of nodes (vertices) V and edges E, denoted $ G(V,E) $
Vertex set on graph G is denoted $ V(G) $
Edge set on graph G is denoted $ E(G) $
Number of vertices in a graph is called the order and is denoted $ |G| $
Number of edges in a graph is denoted $ ||G|| $
Vertex $ v \in V $ and edge $ e \in E $ are incident if the edge connects to the vertex.
Set of all edges at a particular vertex v is denoted $ E(v) $
Two vertices x, y are adjacent on a graph if there is an edge with endpoints x and y
If all vertices are pairwise adjacent, the graph is complete
For two graphs $ G = (V,E) $ and $ G' = (V',E') $, the graphs are isomorphic if there exists a biijection from G to G'.
If we have two graphs G and G', we say that G' is a subgraph of G if all V' subset of V and all E' subset of E
A subgraph G' is a spanning subgraph of G if all V' span all of G (if V' = V)
Degree definitions
Set of neighbors of a vertex v is denoted $ N(v) $
Degree of a vertex v is denoted $ d(v) $ and is equal to the number of edges $ |E(v)| $ at v
Vertex of degree 0 is isolated
The vertex on the graph with the smallest degree $ \delta(G) = \min \left( d(v) | v \in V \right) $ is the minimum degree of G
The vertex on the graph with the largest degree $ \Delta(G) = \max \left( d(v) | v \in V \right) $ is the maximum degree of G
The average degree of G is given by the expression $ d(G) = \dfrac{1}{|V|} \sum_{v \in V} d(v) $
Ratio of edges to vertices on a graph is $ \epsilon(G) = \dfrac{|E|}{|V|} $
If we define edges as having two endpoints, then adding up the degrees of all vertices will lead to twice the number of edges. Mathematically: $ |E| = \dfrac{1}{2} \sum_{v \in V} d(v) = \dfrac{1}{2} d(G) |V| $
This leads to the identity $ \epsilon(G) = \dfrac{1}{2} d(G) $ and the theorem that the number of vertices of odd degree in a graph must always be even. Contrawise proof: if the number of vertices of odd degree is odd, the number of edges is not be an integer.
Path and Cycle Definitions
A path P on a graph G is a non-empty graph that contains vertices and edges that are in G: $ V = \{ x_1, x_2, \dots, x_k\} $ and $ E = \{ x_0 x_1, x_1 x_2, \dots, x_{k-1} x_k \} $
A path is usually referred to by the sequence of vertices it visits, or as a path "from/between x1 to xk"
Independent paths are paths containing no common (internal) vertices. Independent paths may share endpoints though.
We can denote parts of a path using special notation: if a path $ P = x_0 \dots x_{k} $, then the following notation is used to denote only a part of that path:
$ \begin{align} P x_i = x_0 \dots x_i \qquad 0 \leq i \leq k \\ x_i P = x_i \dots x_k \\ x_i P x_j = x_i \dots x_j \end{align} $
We can also connect paths using unions, or by using more shorthand:
$ P x \bigcup x Q y \bigcup y R = P x Q y R $
A cycle C consists of a path whose final edge connects the last node to the first node. Given a path $ P = x_0 \dots x_{k-1} $ the cycle is then $ C = P + x_{k-1} x_0 $
A k-cycle is denoted $ C^k $ and is a cycle of length k.
The girth of a cycle is the number of edges or vertices in a cycle in a graph G. The circumference of a graph is the maximum length of a cycle in a graph G.
The distance of two vertices x and y is the length of the shortest path from x to y $ d_G(x,y) $.
A vertex is central if greatest distance from any other vertex is as small as possible. This minimum distance is the radius of the graph G. Formally:
$ rad(G) = \min_{x \in V(G)} \max_{y \in V(G)} d_G(x,y) $
Note that the radius of a graph is different from the minimum/average degree.
Connectivity
A graph is connected if any two arbitrary vertices are connected.
If the graph is directed, a connected graph means that for any two arbitrary vertices u and v, there is an edge connecting u to v or v to u. A strongly connected graph means that for any two arbitrary nodes u and v, there is an edge connecting u to v and another edge connecting v to u.
Suppose we have two sets of vertices A and B, and a third set of vertices X. Further suppose that any path that connects a vertex from A to a vertex from B contains a vertex from X. Then we say that X separates the vertex sets A and B.
A subgraph of G that is maximally connected (contains every vertex in G) is a component of G. If a component is connected, it is always non empty.
Vertex connectivity: A graph G is k-connected if it has more than k vertices and if no two vertices of G are separated by fewer than k vertices. The maximum value of k such that G is k-connected is the connectivity of G and is denoted $ \kappa(G) $.
Edge connectivity: A graph G is l-edge-connected if every vertex is connected with fewer than l edges (this is a bit unclear). The edge connectivity is denoted $ \lambda(G) $.
Theorem due to Mader 1972: Every graph of average degree at least 4k has a k-connected subgraph. (Can prove inductively.)
Trees and Forests
Acyclic graphs are called forests. Connected forests are called trees.
A connected graph with n vertices is a tree if and only if it has n-1 edges.
We can (but don't have to) pick a particular node to be special - the root of the tree. In that case it is a rooted tree. When we pick a root, this imposes an ordering (assuming vertices can be compared). Given two nodes x and y, we say that $ x \leq y \mbox{ if } x \in rTy $.
A rooted tree T is called normal if any two vertices that are adjacent in the graph are comparable. Every graph has a normal spanning tree.
Bipartite Graphs
A k-partite graph is a graph where the set of vertices V can be partitioned into k classes, such that every edge that starts in one partition will end in a different partition.
If we can select any two vertices from two different classes and they are connected, the k-partite graph is complete.
Bipartite graphs cannot contain cycles of odd lengths. This is always true, so that we can identify bipartite graphs using this property: a graph is bipartite iff it contains no odd cycle.
Edge Contraction
Chapter 2: Matching
Bipartite graph matching
k-partite graph matching
Chapter 3: Connectivity
2-connected graphs
3-connected graphs
Menger's Theorem
Mader's Theorem
Spanning trees (and edge-disjoint spanning trees)
Chapter 4: Planar Graphs
Topology
Plane graphs
Algebraic criteria
Chapter 5: Coloring
Coloring vertices
Coloring edges
Chapter 6: Flows
Circulations
Flows in networks
k-flows
Flow coloring
Tutte's flow conjectures
Chapter 7 and 8: Substructures
Subgraphs
Regularity lemma
Hadwigen's theorem