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A graph with edges colored to illustrate a closed walk, H–A–B–A–H, in green; a circuit which is a closed walk in which all edges are distinct, B–D–E–F–D–C–B, in blue; and a cycle which is a closed walk in which all vertices are distinct, H–D–G–H, in red.
Circular layouts are a good fit for communications network topologies such as star or ring networks, [1] and for the cyclic parts of metabolic networks. [2] For graphs with a known Hamiltonian cycle, a circular layout allows the cycle to be depicted as the circle, and in this way circular layouts form the basis of the LCF notation for Hamiltonian cubic graphs.
The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph ; more generally, intersection graphs of interior-disjoint geometric objects ...
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.
In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The circular chromatic number of a graph G {\displaystyle G} , denoted χ c ( G ) {\displaystyle \chi _{c}(G)} can be given by any of the following definitions, all of which are equivalent (for finite graphs).
A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a chord diagram.That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
The intersection graph of the lines in a hyperbolic arrangement can be an arbitrary circle graph. The corresponding concept to hyperbolic line arrangements for pseudolines is a weak pseudoline arrangement , [ 52 ] a family of curves having the same topological properties as lines [ 53 ] such that any two curves in the family either meet in a ...
Since the graph of is closed, for every point (, ′) on the "vertical line at x", with ′ (), draw an open rectangle ′ ′ disjoint from the graph of . These open rectangles, when projected to the y-axis, cover the y-axis except at f ( x ) {\displaystyle f(x)} , so add one more set V {\displaystyle V} .