What is a minor in a graph?
A minor of a graph G describes a substructure of G that is more general than a subgraph. If we take a subgraph of G and then contract some connected pieces in this subgraph to single points, the re- sulting graph is called a minor of G.
What is the first theorem of graph theory?
The following theorem is often referred to as the First Theorem of Graph The- ory. Theorem 1.1. In a graph G, the sum of the degrees of the vertices is equal to twice the number of edges. Consequently, the number of vertices with odd degree is even.
What is fundamental theorem of graph theory?
In Graph Theory, Handshaking Theorem states in any given graph, Sum of degree of all the vertices is twice the number of edges contained in it.
Is a graph a minor of itself?
Any graph G that can be produced from G by successive application of these reductions is called a minor of G. (In particular, G is a minor of itself.) Every graph that is isomorphic to a minor of G is also called a minor of G. A minor that is not isomorphic to G is called a proper minor.
What is a topological minor?
Topological minors A graph H is called a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G. It is easy to see that every topological minor is also a minor.
Are trees bipartite?
Every tree is bipartite. Cycle graphs with an even number of vertices are bipartite. Every planar graph whose faces all have even length is bipartite.
Is the Petersen graph Hamiltonian?
The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle.
What is c5 in graph theory?
Definition. This undirected graph is defined in the following equivalent ways: It is the cycle graph on 5 vertices, i.e., the graph. It is the Paley graph corresponding to the field of 5 elements. It is the unique (up to graph isomorphism) self-complementary graph on a set of 5 vertices.