Graph Contract For Free

Contraction method: The graph G.e is obtained after the contraction of the edge e of G (See Fig 2). The deletion and the contraction methods are important results in the enumerative combinatorics, that are due to the origin Fussner, which allows the research of the spanning trees of a planar graph. ...

?(G) (using the Greek letter delta) is the maximum degree of a vertex in G, and ?(G) is the minimum degree; see degree. In a graph of n nodes, the density is the ratio of the number of edges of the graph to the number of edges in a complete graph on n nodes.

Star Contraction: One vertex of each component is identified as the center of the star and all other vertices are directly connected to it. Tree Contraction: Generalizing star contraction, disjoint trees within the graph are identified and tree contraction is performed on these trees.

Definition ? A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a represents an endpoint of an edge. An edge joins two vertices a, b and is represented by set of vertices it connects.

A contraction-set E in G = (V,E) is a set of edges E ? E, such that G[E ] is a forest. A contraction H of G is a graph such that there exists a contraction-set E with: H = G/E .

The contraction of a pair of vertices and of a graph, also called vertex identification, is the operation that produces a graph in which the two nodes and are replaced with a single node such that is adjacent to the union of the nodes to which and were originally adjacent.

Definition 1. [3] The corona G ? H of graphs G and H is the graph obtained by taking one copy of G and | V (G) | copies H and then forming the join <v> +Hv = v + Hv for each v ? V (G), where Hv is a copy of H corresponding to the vertex v. V (Hu) ? ?.