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Extending Partial Representations of Circular-Arc Graphs [article]

Jiří Fiala, Ignaz Rutter, Peter Stumpf, Peter Zeman
2021 arXiv   pre-print
We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all variants of Helly circular-arc graphs  ...  We also prove that representation extension problem of unit circular-arc graphs is NP-complete.  ...  We thank Bartosz Walczak for inspiring comments, in particular for his hint to extend Theorem 1 to the case of CAR with distinct endpoints.  ... 
arXiv:2108.13076v1 fatcat:om67almxgzhqdcqumg4ausqixq

Extending Partial Representations of Unit Circular-arc Graphs [article]

Peter Zeman
2017 arXiv   pre-print
In this short note we show that this problem is NP-complete for unit circular-arc graphs.  ...  The partial representation extension problem, introduced by Klavík et al. (2011), generalizes the recognition problem.  ...  A graph is an interval graph if it has an interval representation; see Fig. 1a . Circular-arc Graphs. In a circular-arc representation, the sets R v are arcs of a circle; see Fig 1b.  ... 
arXiv:1706.00928v1 fatcat:3ph7zzzvk5c23dpe3vs3pv7pum

Completing orientations of partially oriented graphs [article]

Joergen Bang-Jensen, J. Huang, Xuding Zhu
2015 arXiv   pre-print
Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable, which generalize a previous result.  ...  Orien- tation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically  ...  4.14 The problem of extending partial proper circular arc representations of proper circular arc graphs is solvable in polynomial time.  ... 
arXiv:1509.01301v1 fatcat:dxvh5b4snrbazckz5j3p3vsvc4

Extending Partial Representations of Circle Graphs [chapter]

Steven Chaplick, Radoslav Fulek, Pavel Klavík
2013 Lecture Notes in Computer Science  
The question is whether one can extend R ′ to a representation R of the entire G, i.e., whether one can draw the remaining chords into a partially pre-drawn representation.  ...  The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle.  ...  This problem is clearly a generalization of partial representation extension since one can describe a partial representation using singleton arcs.  ... 
doi:10.1007/978-3-319-03841-4_12 fatcat:ie7nawmzkba43i3vi3v2t4uy7a

Extending Partial Representations of Circle Graphs [article]

Steven Chaplick, Radoslav Fulek, Pavel Klavík
2017 arXiv   pre-print
The question is whether one can extend R' to a representation R of the entire graph G, i.e., whether one can draw the remaining chords into a partially pre-drawn representation to obtain a representation  ...  The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle.  ...  The third author is supported by CE-ITI (P202/12/G061 of GAČR) and Charles University as GAUK 1334217.  ... 
arXiv:1309.2399v2 fatcat:2ktjxyv4nvgxzab345en6y6sjq

Algorithmic aspects of intersection graphs and representation hypergraphs

Martin Charles Golumbic
1988 Graphs and Combinatorics  
The scope of research that has been going on in this general area extends from the mathematical and algorithmic properties of intersection graphs, to their generalizations and graph parameters motivated  ...  The "grandfather" of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.  ...  Edge Paths Tree graphs intersection Circular-arc Intersection Arcs Circle graphs Proper circular-arc Intersection Arcs Circle graphs Interval graphs Intersection Circle containment  ... 
doi:10.1007/bf01864170 fatcat:hc2zljrxq5hvdejv4d45vlg5wm

Page 8624 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
This class of graphs admits some interesting subclasses: proper circular-arc graphs, unit circular- arc graphs, Helly circular-arc graphs and clique-Helly circular- are graphs.  ...  Summary: “The intersection graph of a family of arcs on a cir- cle is called a circular-are graph.  ... 

Obstructions to chordal circular-arc graphs of small independence number

Mathew Francis, Pavol Hell, Juraj Stacho
2013 Electronic Notes in Discrete Mathematics  
In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.  ...  Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph.  ...  . , A k−1 of T (considered as a digraph) where A i are nodes of T . Using this tour, we construct a circular-arc representation of G.  ... 
doi:10.1016/j.endm.2013.10.012 fatcat:xkph4nvkdfenvf437g4jktrnw4

Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems [article]

Thomas Bläsius, Ignaz Rutter
2011 arXiv   pre-print
used to also solve the problem of extending partial interval representations of graphs with n vertices and m edges in time O(n + m), improving a recent result of Klav\'ik et al.  ...  Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child  ...  Extending Partial Interval Representations Let G be a graph, H = (V, E) be a subgraph of G and let I be an interval representation of H.  ... 
arXiv:1112.0245v1 fatcat:s6nsmdywkvhxvjpwgvbn3wdlve

Precoloring extension. I. Interval graphs

M. Biró, M. Hujter, Zs. Tuza
1992 Discrete Mathematics  
Suppose that some vertices of a graph G are assigned to some colors. Can this 'precoloring' be extended to a proper coloring of G with at most k colors (for some given k)?  ...  Interval graphs, Discrete Mathematics 100 (1992) 267-279. This paper is the first article in a series devoted to the study of the following general problem on vertex colorings of graphs.  ...  is NP-complete on circular-arc graphs [7] . Let us consider an arbitrary circular-arc graph G = (V, E) (together with its arc-representation) and arbitrary integer k.  ... 
doi:10.1016/0012-365x(92)90646-w fatcat:hd2t6a6sirdffcexybhw33uloy

Outside-Obstacle Representations with All Vertices on the Outer Face [article]

Oksana Firman, Philipp Kindermann, Jonathan Klawitter, Boris Klemz, Felix Klesen, Alexander Wolff
2022 arXiv   pre-print
An obstacle representation of a graph G consists of a set of polygonal obstacles and a drawing of G as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line  ...  It is known that every outerplanar graph admits such a representation [Alpert, Koch, Laison; DCG 2010]. We strengthen this result by showing that every (partial) 2-tree has an OOR.  ...  Outside-Obstacle Representations for Partial 2-Trees The graph class of 2-trees is recursively defined as follows: K 3 is a 2-tree.  ... 
arXiv:2202.13015v3 fatcat:5teo2yhem5bs7buhc5uvhvg5fu

On powers of m-trapezoid graphs

Carsten Flotow
1995 Discrete Applied Mathematics  
First a new class of graphs is introduced: m-trapezoid graphs are the intersection graphs of m-trapezoids, where an m-trapezoid is given by m + 1 intervals on m + 1 parallel lines.  ...  The main result of this paper is that if GL-' is an m-trapezoid graph then Gk is also an m-trapezoid graph.  ...  I also thank Andreas Parra for the tea-time at which he had told me about trapezoid graphs.  ... 
doi:10.1016/0166-218x(95)00062-v fatcat:g4ig7xejpngpdobmg4vzj2tpve

Contact Representations of Sparse Planar Graphs [article]

Md. Jawaherul Alam, David Eppstein, Michael Kaufmann, Stephen G. Kobourov, Sergey Pupyrev, Andre Schulz, Torsten Ueckerdt
2015 arXiv   pre-print
We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint  ...  We partially answer this question by computing CCA-representations for several subclasses of planar (2,0)-sparse graphs.  ...  We partially answer these questions by computing circular-arc contact representation for several subclasses of planar (2, 0)-sparse graphs, and by finding a (2, 0)-sparse plane multigraph that does not  ... 
arXiv:1501.00318v1 fatcat:lmnfvgc3xbfqbdyq6hwn6hxbsa

Algorithm for the Vertex Connectivity Problem on Circular Trapezoid Graphs

Hirotoshi Honma, Kento Nishimura, Yuto Tamori, Yoko Nakajima
2019 Journal of Applied Mathematics and Physics  
The vertex connectivity ( ) G κ of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory.  ...  In this paper, we designed an ( ) 2 O n time algorithm to solve connectivity problem on circular trapezoid graphs.  ...  This work was partially supported by JSPS KAKENHI Grant Number 19K11834 and 17K00324. Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.  ... 
doi:10.4236/jamp.2019.711177 fatcat:yqtjuvmys5a7pncxjwmkqnn7j4

Page 5934 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
Further, arc orders are path orders and path orders are circular.  ...  The authors describe various methods of constructing cyclic orders: poset-generated C (X is partially ordered and (x,y,z) €T if and only if either x << y<zory<z< xX or z<x<y), arc order C (elements are  ... 
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