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### Intersection Graphs of Non-crossing Paths [article]

Steven Chaplick
2020 arXiv   pre-print
We study 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] .  ...

### Non-crossing Paths with Geographic Constraints [chapter]

Rodrigo I. Silveira, Bettina Speckmann, Kevin Verbeek
2018 Lecture Notes in Computer Science
We prove that when 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.  ...

### Non-crossing paths with geographic constraints [article]

Rodrigo I. Silveira, Bettina Speckmann, Kevin Verbeek
2019 arXiv   pre-print
We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs 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.  ...

### Graphs of edge-intersecting and non-splitting paths

Arman Boyacı, Tınaz Ekim, Mordechai Shalom, Shmuel Zaks
2016 Theoretical Computer Science
Recently we introduced the class 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 ′ .  ...

### Shortest non-trivial cycles in directed surface graphs

Jeff Erickson
2011 Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11
Let G be a directed 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  ...

### Triangle-Free Planar Graphs as Segment Intersection Graphs [chapter]

Natalia de Castro, Francisco Javier Cobos, Juan Carlos Dana, Alberto Márquez, Marc Noy
2004 Graph Algorithms and Applications 3
We prove that every triangle-free planar 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.  ...

### Algorithms for finding non-crossing paths with minimum total length in plane graphs [chapter]

Jun-ya Takahashi, Hitoshi Suzuki, Takao Nishizeki
1992 Lecture Notes in Computer Science
Here "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.  ...

### Algorithms for finding noncrossing paths with minimum total length in plane graphs

Jun-Ya Takahashi, Hitoshi Suzuki, Takao Nishizeki
1995 Electronics and communications in Japan. Part 3, Fundamental electronic science
Here "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.  ...

### Intersecting longest paths in chordal graphs [article]

Daniel J. Harvey, Michael S. Payne
2020 arXiv   pre-print
We consider the size 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 .  ...

### Graphs of Triangulations and Perfect Matchings

M.E. Houle, F. Hurtado, M. Noy, E. Rivera-Campo
2005 Graphs and Combinatorics
is a single 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  ...

### Edge-intersection graphs of grid paths: The bend-number

Daniel Heldt, Kolja Knauer, Torsten Ueckerdt
2014 Discrete Applied Mathematics
We investigate edge-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.  ...

### Edge-intersection graphs of grid paths: the bend-number [article]

Daniel Heldt, Kolja Knauer, Torsten Ueckerdt
2012 arXiv   pre-print
We investigate edge-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.  ...

### On Polyhedral Embeddings of Cubic Graphs

BOJAN MOHAR, ANDREJ VODOPIVEC
2006 Combinatorics, probability & computing
On the other hand, for every nonorientable surface S, there exists a 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.  ...

### Ramsey-Type Results for Geometric Graphs, I

G. Károlyi, J. Pach, G. Tóth
1997 Discrete & Computational Geometry
For any 2-coloring 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.  ...

### Non-crossing Connectors in the Plane [chapter]

Jan Kratochvíl, Torsten Ueckerdt
2013 Lecture Notes in Computer Science
We prove that 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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