Circle Graph: Definition and Examples is a video and lesson on circle graphs.
A circle graph is an intersection graph of a set of circles.It is an undirected graph that can be associated with a circle if the corresponding chords cross each other.
If a given n-vertex undirected graph is a circle graph, it can be tested by an O(n2)-time algorithm.
When restricted to circle graphs, a number of NP-complete problems have polynomial time algorithms.An optimal tree decomposition can be constructed in O(n 3) time if the treewidth of a circle graph is determined.A minimum fill-in with as few edges as possible may be found in O(n3) time.A maximum independent set of an unweighted circle graph can be found in O(n log2 n) time, according to Tiskin and Nash and Gregg.
There are problems that remain NP- complete when restricted to circle graphs.Minimum total dominating set problems are included.[3]
The minimum number of colors that can be used to color a circle graph is called the chromatic number.Since it is possible to form circle graphs in which arbitrarily large sets of chords all cross each other, the number of a circle graph is NP- complete.The test is to see if a circle graph can be colored by four colors.There are many details missing from the writeup of the result of finding a coloring with three colors.[5]
Problems of coloring restricted subclasses of circle graphs with few colors have been investigated by several authors.It is possible to color a circle graph with as few as 7 k 2 displaystyle 7k2 colors.One way of saying this is that the circle graphs are bound.All triangle-free circle graphs can be colored with five colors if k is at most five.If a circle graph is triangle-free and has no four-vertex cycles, it can be colored with most three colors.In this correspondence, the number of colors in the coloring corresponds to the amount of trees in a product representation.10
Circle graphs are an abstract representation of a special case for wire routing.All nets are two-terminal, and the terminals are placed on the perimeter of the rectangle.The intersection of the nets is a circle graph.Ensuring that different nets stay disconnected and their potential intersecting parts are laid out in different conducting layers are among the goals of the wire route step.Various aspects of the problem are captured in circle graphs.
If the edges of a given graph G are arranged on a circle with an intersection between them, then the circle graph and colorings of this can be used to find a book embedded in a graph.The number of colors in the coloring corresponds to the amount of pages in a book.[5]