Complement graph

In the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there. The complement is not the set complement of the graph; only the edges are complemented. Let G = (V, E) be a simple graph and let K consist of all 2-element subsets of V. Then H = (V, K \ E) is the complement of G, where K \ E is the relative complement of E in K. For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of V in place of the set K in the formula above. In terms of the adjacency matrix A of the graph, if Q is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries are unity except the diagonal entries which are zero), then the adjacency matrix of the complement of A is Q-A. The complement is not defined for multigraphs. In graphs that allow self-loops (but not multiple adjacencies) the complement of G may be defined by adding a self-loop to every vertex that does not have one in G, and otherwise using the same formula as above. This operation is, however, different from the one for simple graphs, since applying it to a graph with no self-loops would result in a graph with self-loops on all vertices. Several graph-theoretic concepts are related to each other via complementation: The complement of an edgeless graph is a complete graph and vice versa. Any induced subgraph of the complement graph of a graph G is the complement of the corresponding induced subgraph in G. An independent set in a graph is a clique in the complement graph and vice versa. This is a special case of the previous two properties, as an independent set is an edgeless induced subgraph and a clique is a complete induced subgraph.

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Induced subgraph

In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges (from the original graph) connecting pairs of vertices in that subset. Formally, let be any graph, and let be any subset of vertices of G. Then the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in . That is, for any two vertices , and are adjacent in if and only if they are adjacent in .

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In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. The neighbourhood is often denoted N_G (v) or (when the graph is unambiguous) N(v). The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs.

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