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For instance, if G and H are both connected graphs, each having at least four vertices and having exactly twice as many total vertices as their domination numbers, then γ(G H) = γ(G) γ(H). [2] The graphs G and H with this property consist of the four-vertex cycle C 4 together with the rooted products of a connected graph and a single edge. [2]
In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H, and that has as many vertices as possible. Finding this graph is NP-hard. In the associated decision problem, the input is two graphs G and H and a number k.
Subgraph isomorphism is a generalization of the graph isomorphism problem, which asks whether G is isomorphic to H: the answer to the graph isomorphism problem is true if and only if G and H both have the same numbers of vertices and edges and the subgraph isomorphism problem for G and H is true. However the complexity-theoretic status of graph ...
2. The Frucht graph, one of the two smallest cubic graphs with no nontrivial symmetries. 3. Frucht's theorem that every finite group is the group of symmetries of a finite graph. full Synonym for induced. functional graph A functional graph is a directed graph where every vertex has out-degree one. Equivalently, a functional graph is a maximal ...
The case of exact graph matching is known as the graph isomorphism problem. [1] The problem of exact matching of a graph to a part of another graph is called subgraph isomorphism problem. Inexact graph matching refers to matching problems when exact matching is impossible, e.g., when the number of vertices in the two graphs are different. In ...
Two graphs that have the same deck are said to be hypomorphic. With these definitions, the conjecture can be stated as: Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic. (The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)
Clique-sums have a close connection with treewidth: If two graphs have treewidth at most k, so does their k-clique-sum.Every tree is the 1-clique-sum of its edges. Every series–parallel graph, or more generally every graph with treewidth at most two, may be formed as a 2-clique-sum of triangles.
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...