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Intersection Graphs of Non-crossing Paths
[article]

2020
*
arXiv
*
pre-print

We study

arXiv:1907.00272v2
fatcat:paeipvyuk5gmxl6wcunfatj56a
*graph*classes modeled by families*of**non*-*crossing*(NC) connected sets. Two classic*graph*classes in this context are disk*graphs*and proper interval*graphs*. ... For the*intersection**graphs**of*NC*paths**of*a tree, we characterize the minimum connected dominating sets (leading to a linear time algorithm to compute one). ... The most general case*of**intersection**graphs**of**non*-*crossing*sets which has been studied is the class*of**intersection**graphs**of**non*-*crossing*connected (NC-C) sets in the plane [41] . ...##
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Non-crossing Paths with Geographic Constraints
[chapter]

2018
*
Lecture Notes in Computer Science
*

We prove that when

doi:10.1007/978-3-319-73915-1_35
fatcat:2jsfau5vsbfphhl7x5jnmymiem
*paths*must be drawn as straight line segments, it is NP-complete to determine if a*crossing*-free solution exists. ... We focus on the seemingly simple setting where each region is a unit length vertical segment, and one wants to connect pairs*of*segments with a*path*that lies inside the convex hull*of*the two segments ... The key property*of*the clause gadget is that there exist*non*-*crossing**paths*connecting the three tubes if and only if at least one literal is true. ...##
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Non-crossing paths with geographic constraints
[article]

2019
*
arXiv
*
pre-print

We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs

arXiv:1708.05486v3
fatcat:oflzdcw5lvbvve3y4axeyy725q
*of*segments with a*path*that lies inside the convex hull*of*the two segments. ... In the more general case*of**paths*that can have any shape, we show that the problem is polynomial under certain assumptions. ... The key property*of*the clause gadget is that there exist*non*-*crossing**paths*connecting the three tubes if and only if at least one literal is true. ...##
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Graphs of edge-intersecting and non-splitting paths

2016
*
Theoretical Computer Science
*

Recently we introduced the class

doi:10.1016/j.tcs.2015.10.004
fatcat:4r5rnxg64bethnioy5wu2ip22i
*of**graphs**of*Edge-*Intersecting*and*Non*-Splitting*Paths*in a Tree (ENPT) [2] . ... We also show that the class ENP coincides with the family*of**graphs**of*Edge-*Intersecting*and*Non*-Splitting*Paths*in a Grid (ENPG). ...*Graphs**of*edge-*intersecting*and*non*-splitting*paths*ii) split(P, P ′ ) corresponds to the set*of*all*non*-terminating segment endpoints*crossed*by both P and P ′ . ...##
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Shortest non-trivial cycles in directed surface graphs

2011
*
Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11
*

Let G be a directed

doi:10.1145/1998196.1998231
dblp:conf/compgeom/Erickson11
fatcat:zndqeloqv5b3fa5h6da6ns2om4
*graph*embedded on a surface*of*genus g. ... We also describe an algorithm to compute the shortest*non*-contractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected*graphs**of*constant genus. ... Our key observation is that although the shortest*non*-contractible cycle γ may*intersect*each*of*these shortest*paths*arbitrarily many times, at most one*intersection*with any shortest*path*is topologically ...##
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Triangle-Free Planar Graphs as Segment Intersection Graphs
[chapter]

2004
*
Graph Algorithms and Applications 3
*

We prove that every triangle-free planar

doi:10.1142/9789812796608_0002
fatcat:k2n5jjby5nanpakg5yhnacgd7a
*graph*is the*intersection**graph**of*a set*of*segments in the plane. ... This particular class*of**intersection**graphs*is also known as contact*graphs*. ... by*non*-*crossing*segments. ...##
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Algorithms for finding non-crossing paths with minimum total length in plane graphs
[chapter]

1992
*
Lecture Notes in Computer Science
*

Here "

doi:10.1007/3-540-56279-6_92
fatcat:u4evppy5b5cijlvg4ax2jfear4
*non*-*crossing**paths*" may share common vertices or edges but do not*cross*each other in the plane. The algorithm runs in time O(nlogn) where n is the number*of*vertices in G. ... This paper presents an algorithm for finding k "*non*-*crossing**paths*" in G, each connecting a terminal pair, whose total length is minimum. ... This research is partly supported by Grant in Aid for Scientific Research*of*the Ministry*of*Education, Science, and Culture*of*Japan under a grant number: General Research (C) 04650300. ...##
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Algorithms for finding noncrossing paths with minimum total length in plane graphs

1995
*
Electronics and communications in Japan. Part 3, Fundamental electronic science
*

Here "

doi:10.1002/ecjc.4430780401
fatcat:q5yicvzyh5aaza6tiu7zy5lffy
*non*-*crossing**paths*" may share common vertices or edges but do not*cross*each other in the plane. The algorithm runs in time O(nlogn) where n is the number*of*vertices in G. ... This paper presents an algorithm for finding k "*non*-*crossing**paths*" in G, each connecting a terminal pair, whose total length is minimum. ... This research is partly supported by Grant in Aid for Scientific Research*of*the Ministry*of*Education, Science, and Culture*of*Japan under a grant number: General Research (C) 04650300. ...##
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Intersecting longest paths in chordal graphs
[article]

2020
*
arXiv
*
pre-print

We consider the size

arXiv:2012.07221v1
fatcat:ukzjls5az5ajxl3fbzysj4xrfy
*of*the smallest set*of*vertices required to*intersect*every longest*path*in a chordal*graph*. Such sets are known as longest*path*transversals. ... We also consider the analogous question for longest cycles, and show that if G is a 2-connected chordal*graph*then there is a transversal*intersecting*all longest cycles*of*order at most 2⌈ω(G)/3⌉. ... We construct a*path*P M as follows: • Consider the*graph*G − N and the*path*P inside this*graph*. Since P*intersects*N , G − N will not contain all*of*P . ...##
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Graphs of Triangulations and Perfect Matchings

2005
*
Graphs and Combinatorics
*

is a single

doi:10.1007/s00373-005-0615-2
fatcat:cpwoct2ksndx5ghgm4qfwlbg4m
*non*-*crossing*cycle, is also connected. ... A main tool in our proof is a result*of*independent interest, namely that the*graph*M(P ) that has as vertices the*non*-*crossing*perfect matchings*of*P and two*of*them are adjacent if their symmetric difference ... The*graph**of**non*-*crossing*perfect matchings M(P )*of*P is the*graph*with one vertex for each*non*-*crossing*perfect matching*of*P , in which two matchings are adjacent if and only if one can be obtained ...##
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Edge-intersection graphs of grid paths: The bend-number

2014
*
Discrete Applied Mathematics
*

We investigate edge-

doi:10.1016/j.dam.2013.10.035
fatcat:kgcvwsfeibfsxbhfvg4ec56yim
*intersection**graphs**of**paths*in the plane grid, regarding a parameter called the bend-number. ... The bend-number is the minimum k such that grid-*paths*with at most k bends each suffice to represent a given*graph*. This parameter is related to the interval-number and the track-number*of*a*graph*. ... In particular two*paths*representing*non*-adjacent vertices may*intersect*in grid-points. ...##
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Edge-intersection graphs of grid paths: the bend-number
[article]

2012
*
arXiv
*
pre-print

We investigate edge-

arXiv:1009.2861v3
fatcat:rg2hpnpbrncbbjzpqqornusgsm
*intersection**graphs**of**paths*in the plane grid, regarding a parameter called the bend-number. ... The bend-number is the minimum k such that grid-*paths*with at most k bends each suffice to represent a given*graph*. This parameter is related to the interval-number and the track-number*of*a*graph*. ... In particular two*paths*representing*non*-adjacent vertices may*intersect*in grid-points. ...##
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On Polyhedral Embeddings of Cubic Graphs

2006
*
Combinatorics, probability & computing
*

On the other hand, for every nonorientable surface S, there exists a

doi:10.1017/s0963548306007607
fatcat:m2hxp427pnetjpbk54tipe222a
*non*3-edge-colorable*graph*, which polyhedrally embeds in S. ... Polyhedral embeddings*of*cubic*graphs*by means*of*certain operations are studied. It is proved that some known families*of*snarks have no (orientable) polyhedral embeddings. ... These cycles are all*cross*faces. As in the proof*of*Lemma 4.4, we see that there are at least four*intersections**of**cross*faces. ...##
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Ramsey-Type Results for Geometric Graphs, I

1997
*
Discrete & Computational Geometry
*

For any 2-coloring

doi:10.1007/pl00009317
fatcat:kopc2u4aczaqfaq3rjrjwd6rvq
*of*the n 2 segments determined by n points in general position in the plane, at least one*of*the color classes contains a*non*-self-*intersecting*spanning tree. ...*crossing*. ...*Non*-Self-*Intersecting**Paths*. The length*of*a*path*is the number*of*its edges. Let P n denote the class*of*all*non*-self-*intersecting**paths**of*length n. ...##
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Non-crossing Connectors in the Plane
[chapter]

2013
*
Lecture Notes in Computer Science
*

We prove that

doi:10.1007/978-3-642-38236-9_11
fatcat:fxkt6wmmbvgftlfz7iscae32se
*non*-*crossing*connectors do always exist if the regions form a collection*of*pseudo-disks, i.e., the boundaries*of*every pair*of*regions*intersect*at most twice. ... We consider the*non*-*crossing*connectors problem, which is stated as follows: Given n regions R1, . . . , Rn in the plane and finite point sets Pi ⊂ Ri for i = 1, . . . , n, are there*non*-*crossing*connectors ... Sam Loyds Cyclopedia*of*5000 puzzles, tricks and conundrums published in 1914 refers to a puzzle*of*connecting houses to gates by*non*-*crossing**paths*. ...
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