![]() Among the goals of wire routing step is to ensure that different nets stay electrically disconnected, and their potential intersecting parts must be laid out in different conducting layers. It is easily seen that the intersection graph of these nets is a circle graph. ![]() In this case the routing area is a rectangle, all nets are two-terminal, and the terminals are placed on the perimeter of the rectangle. Applications Ĭircle graphs arise in VLSI physical design as an abstract representation for a special case for wire routing, known as "two-terminal switchbox routing". The problem of coloring triangle-free squaregraphs is equivalent to the problem of representing squaregraphs as isometric subgraphs of Cartesian products of trees in this correspondence, the number of colors in the coloring corresponds to the number of trees in the product representation. If a circle graph has girth at least five (that is, it is triangle-free and has no four-vertex cycles) it can be colored with at most three colors. In the particular case when k = 3 (that is, for triangle-free circle graphs) the chromatic number is at most five, and this is tight: all triangle-free circle graphs may be colored with five colors, and there exist triangle-free circle graphs that require five colors. Nash & Gregg (2010) have shown that a maximum independent set of an unweighted circle graph can be found in O( n min -bounded. Tiskin (2010) has shown that a maximum clique of a circle graph can be found in O( n log 2 n) time, while Additionally, a minimum fill-in (that is, a chordal graph with as few edges as possible that contains the given circle graph as a subgraph) may be found in O( n 3) time. For instance, Kloks (1996) showed that the treewidth of a circle graph can be determined, and an optimal tree decomposition constructed, in O( n 3) time. Spinrad (1994) gives an O( n 2)-time algorithm that tests whether a given n-vertex undirected graph is a circle graph and, if it is, constructs a set of chords that represents it.Ī number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs. ![]() 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. In graph theory, a circle graph is the intersection graph of a chord diagram. A circle with five chords and the corresponding circle graph. ![]()
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