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In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn.
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
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:
The line graphs of cubic graphs are 4-regular and have a Hamiltonian decomposition if and only if the underlying cubic graph has a Hamiltonian cycle. [12] [13] As a consequence, Hamiltonian decomposition remains NP-complete for classes of graphs that include line graphs of hard instances of the Hamiltonian cycle problem. For instance ...
quasi-line graph A quasi-line graph or locally co-bipartite graph is a graph in which the open neighborhood of every vertex can be partitioned into two cliques. These graphs are always claw-free and they include as a special case the line graphs. They are used in the structure theory of claw-free graphs. quasi-random graph sequence
A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle , Hamiltonian circuit , vertex tour or graph cycle is a cycle that visits each vertex exactly once.
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