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Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and is therefore NP-complete. [1] However certain other cases of subgraph isomorphism may be solved in polynomial time. [2] Sometimes the name subgraph matching is also used for the same problem ...
Every graph contains at most 3 n/3 maximal independent sets, [5] but many graphs have far fewer. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence. [6] Therefore, both numbers are proportional to ...
The yellow directed acyclic graph is the condensation of the blue directed graph. It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex. If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of G.
The achromatic number of a graph is the maximum number of colors in a complete coloring. 5. A complete invariant of a graph is a synonym for a canonical form, an invariant that has different values for non-isomorphic graphs. component A connected component of a graph is a maximal connected subgraph.
The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G [ S ] {\displaystyle G[S]} may also be called the subgraph induced in G {\displaystyle G} by S {\displaystyle S} , or (if context makes the choice of G {\displaystyle G} unambiguous) the induced subgraph of S {\displaystyle S} .
The number of perfect matchings in a complete graph K n (with n even) is given by the double factorial (n − 1)!!. [13] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are given by the telephone numbers. [14] The number of perfect matchings in a graph is also known as the hafnian of its adjacency ...
The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset. [2]
Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph. [4] [5] In any graph other than one formed by adding one edge to a cycle, a peripheral cycle must be an induced cycle.