scholarly journals Isotropic Matroids II: Circle Graphs

10.37236/5223 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Circle Graph ◽  
Circle Graphs

We present several characterizations of circle graphs, which follow from Bouchet’s circle graph obstructions theorem.

scholarly journals A Characterization of Circle Graphs in Terms of Multimatroid Representations

10.37236/6992 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Robert Brijder ◽  
Lorenzo Traldi
Keyword(s):  
Simple Graph ◽  
Rank Function ◽  
Binary Matroid ◽  
Circle Graph ◽  
Circle Graphs ◽  
Isotropic System

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

scholarly journals On a conjecture concerning helly circle graphs

2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Systems Of Equations ◽  
Intersection Graphs ◽  
Circle Graph ◽  
Circle Graphs ◽  
Analytic Tool ◽  
Two Systems ◽  
Cartesian Plane ◽  
Definition Of ◽  
Straight Lines

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.

TREEWIDTH OF CIRCLE GRAPHS

1996 ◽  
Vol 07 (02) ◽  
pp. 111-120 ◽  
Author(s):  
T. KLOKS
Keyword(s):  
Polynomial Time ◽  
Clique Number ◽  
Intersection Graph ◽  
Finite Collection ◽  
Circle Graph ◽  
Circle Graphs

In this paper we show that the treewidth of a circle graph can be computed in polynomial time. A circle graph is a graph that is isomorphic to the intersection graph of a finite collection of chords of a circle. The TREEWIDTH problem can be viewed upon as the problem of finding a chordal embedding of the graph that minimizes the clique number. Our algorithm to determine the treewidth of a circle graph can be implemented to run in O(n3) time, where n is the number of vertices of the graph.

scholarly journals New clique and independent set algorithms for circle graphs

1992 ◽  
Vol 36 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Alberto Apostolico ◽  
Mikhail J. Atallah ◽  
Susanne E. Hambrusch
Keyword(s):  
Independent Set ◽  
Circle Graphs

MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

2011 ◽  
Vol 03 (03) ◽  
pp. 323-336 ◽  
Author(s):  
FANICA GAVRIL
Keyword(s):  
Minimum Weight ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Circle Graphs ◽  
Intersection Model ◽  
The Family ◽  
Positive Weights ◽  
Vertex Set ◽  
Vertex Sets ◽  
Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

scholarly journals Extending Partial Representations of Circle Graphs

2013 ◽  
pp. 131-142 ◽  
Author(s):  
Steven Chaplick ◽  
Radoslav Fulek ◽  
Pavel Klavík
Keyword(s):  
Circle Graphs
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