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Extending Partial Representations of Circular-Arc Graphs
[article]

2021
*
arXiv
*
pre-print

We complement this hardness argument with tractability results

arXiv:2108.13076v1
fatcat:om67almxgzhqdcqumg4ausqixq
*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. ...##
###
Extending Partial Representations of Unit Circular-arc Graphs
[article]

2017
*
arXiv
*
pre-print

In this short note we show that this problem is NP-complete for unit

arXiv:1706.00928v1
fatcat:3ph7zzzvk5c23dpe3vs3pv7pum
*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. ...##
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Completing orientations of partially oriented graphs
[article]

2015
*
arXiv
*
pre-print

Our results imply that the

arXiv:1509.01301v1
fatcat:dxvh5b4snrbazckz5j3p3vsvc4
*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. ...##
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Extending Partial Representations of Circle Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

The question is whether one can

doi:10.1007/978-3-319-03841-4_12
fatcat:ie7nawmzkba43i3vi3v2t4uy7a
*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*. ...##
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Extending Partial Representations of Circle Graphs
[article]

2017
*
arXiv
*
pre-print

The question is whether one can

arXiv:1309.2399v2
fatcat:2ktjxyv4nvgxzab345en6y6sjq
*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. ...##
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Algorithmic aspects of intersection graphs and representation hypergraphs

1988
*
Graphs and Combinatorics
*

The scope

doi:10.1007/bf01864170
fatcat:hc2zljrxq5hvdejv4d45vlg5wm
*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 ...##
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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*. ...##
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Obstructions to chordal circular-arc graphs of small independence number

2013
*
Electronic Notes in Discrete Mathematics
*

In our proof we use a novel geometric approach, constructing a

doi:10.1016/j.endm.2013.10.012
fatcat:xkph4nvkdfenvf437g4jktrnw4
*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. ...##
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Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems
[article]

2011
*
arXiv
*
pre-print

used to also solve the problem

arXiv:1112.0245v1
fatcat:s6nsmdywkvhxvjpwgvbn3wdlve
*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. ...##
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Precoloring extension. I. Interval graphs

1992
*
Discrete Mathematics
*

Suppose that some vertices

doi:10.1016/0012-365x(92)90646-w
fatcat:hd2t6a6sirdffcexybhw33uloy
*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. ...##
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Outside-Obstacle Representations with All Vertices on the Outer Face
[article]

2022
*
arXiv
*
pre-print

An obstacle

arXiv:2202.13015v3
fatcat:5teo2yhem5bs7buhc5uvhvg5fu
*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. ...##
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On powers of m-trapezoid graphs

1995
*
Discrete Applied Mathematics
*

First a new class

doi:10.1016/0166-218x(95)00062-v
fatcat:g4ig7xejpngpdobmg4vzj2tpve
*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*. ...##
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Contact Representations of Sparse Planar Graphs
[article]

2015
*
arXiv
*
pre-print

We study

arXiv:1501.00318v1
fatcat:lmnfvgc3xbfqbdyq6hwn6hxbsa
*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 ...##
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Algorithm for the Vertex Connectivity Problem on Circular Trapezoid Graphs

2019
*
Journal of Applied Mathematics and Physics
*

The vertex connectivity ( ) G κ

doi:10.4236/jamp.2019.711177
fatcat:yqtjuvmys5a7pncxjwmkqnn7j4
*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. ...##
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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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