Search results
Results from the WOW.Com Content Network
The line graph of a graph G is defined as the intersection graph of the edges of G, where we represent each edge as the set of its two endpoints. A string graph is the intersection graph of curves on a plane. A graph has boxicity k if it is the intersection graph of multidimensional boxes of dimension k, but not of any smaller dimension.
In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis , following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976).
In the mathematical field of graph theory, the intersection number of a graph = (,) is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element.
Klein & Margraf (2003) define the linear intersection number of a graph, similarly, to be the minimum number of vertices in a linear hypergraph whose line graph is G. As they observe, the Erdős–Faber–Lovász conjecture is equivalent to the statement that the chromatic number of any graph is at most equal to its linear intersection number.
In graph theory, particularly in the theory of hypergraphs, the line graph of a hypergraph H, denoted L(H), is the graph whose vertex set is the set of the hyperedges of H, with two vertices adjacent in L(H) when their corresponding hyperedges have a nonempty intersection in H. In other words, L(H) is the intersection graph of a family of ...
A result of de Castro et al. (2002) combines Grötzsch's theorem with Scheinerman's conjecture on the representation of planar graphs as intersection graphs of line segments. They proved that every triangle-free planar graph can be represented by a collection of line segments, with three slopes, such that two vertices of the graph are adjacent ...
An intersection graph is a graph whose vertices correspond to sets or geometric objects, with an edge between two vertices exactly when the corresponding two sets or objects have a nonempty intersection. Several classes of graphs may be defined as the intersection graphs of certain types of objects, for instance chordal graphs (intersection ...
graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets: