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Drawing Shortest Paths in Geodetic Graphs
[article]

2020
*
arXiv
*
pre-print

Motivated by the fact that

arXiv:2008.07637v1
fatcat:pp7zfwmd4jcjbofvnh44fkl53e
*in*a space where*shortest**paths*are unique, no two*shortest**paths*meet twice, we study a question posed by Greg Bodwin: Given a*geodetic**graph*G, i.e., an unweighted*graph**in*... which the*shortest**path*between any pair of vertices is unique, is there a philogeodetic*drawing*of G, i.e., a*drawing*of G*in*which the curves of any two*shortest**paths*meet at most once? ... Observe that any two*shortest**paths**in*a*geodetic**graph*are either disjoint or they intersect*in*a*path*. Thus, a planar*drawing*of a planar*geodetic**graph*is philogeodetic. ...##
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Hardness and approximation for the geodetic set problem in some graph classes
[article]

2019
*
arXiv
*
pre-print

A set of vertices S of a

arXiv:1909.08795v1
fatcat:fpe4dfo6l5f4xltxjwescdepk4
*graph*G is a*geodetic*set if any vertex of G lies*in*some*shortest**path*between some pair of vertices from S. ...*In*this paper, we study the computational complexity of finding the*geodetic*number of*graphs*. ... First, we observe that,*in*any vertex gadget G v that is part of f (G), the unique*shortest**path*between two distinct vertices Hardness and approximation for the*geodetic*set problem*in*some*graph*classes ...##
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Strong geodetic problems in networks

2018
*
Discussiones Mathematicae Graph Theory
*

The strong

doi:10.7151/dmgt.2139
fatcat:pavbtvghfra2fcxw52eye6dvyq
*geodetic*number is compared with the*geodetic*number and with the isometric*path*number. It is determined for several families of*graphs*including Apollonian networks. ... Applying Sierpiński*graphs*, an algorithm is developed that returns a minimum*path*cover of Apollonian networks corresponding to the strong*geodetic*number. ...*In*particular, the present short proof of Proposition 3.1 and the recursive algorithm from the end of Section 3 were proposed to us by one of the referees. ...##
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Algorithms and complexity for geodetic sets on planar and chordal graphs
[article]

2020
*
arXiv
*
pre-print

A set S of vertices of a

arXiv:2006.16511v1
fatcat:ax56pso6ynb4bmc3jb2yf6rt6i
*graph*G is a*geodetic*set if every vertex of G lies*in*a*shortest**path*between some pair of vertices of S. ... The Minimum*Geodetic*Set (MGS) problem is to find a*geodetic*set with minimum cardinality of a given*graph*. ... However, the case k = 2 (i.e.*graphs*of tree-width 2) remains open. ...##
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Strong geodetic problem in networks: computational complexity and solution for Apollonian networks
[article]

2017
*
arXiv
*
pre-print

The

arXiv:1708.03868v1
fatcat:3dz6zcra4bcffi47zqfaplosgm
*geodetic*problem was introduced by Harary et al.*In*order to model some social network problems, a similar problem is introduced*in*this paper and named the strong*geodetic*problem. ... It is also proved that*in*general the strong*geodetic*problem is NP-complete. ... Let G = (V, E) be the*graph*corresponding to the social network. If S ⊆ V , then for each pair of vertices x, y ∈ S, x = y, let g(x, y) be a selected fixed*shortest*x, y-*path*. ...##
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Page 6732 of Mathematical Reviews Vol. , Issue 2004i
[page]

2004
*
Mathematical Reviews
*

The dis- tance d(u,v) from a vertex uw to a vertex v

*in*an oriented*graph*D is the length of a*shortest*directed u-v*path**in*D. ... The closed interval /[w,v] consists of u and v together with all vertices lying*in*a u-v or v-u*shortest**path*. ...##
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An O(mn^2) algorithm for computing the strong geodetic number in outerplanar graphs

2020
*
Discussiones Mathematicae Graph Theory
*

*In*this paper we show that the strong

*geodetic*number of an outerplanar

*graph*can be computed

*in*polynomial time. ... It is known that it is NP-hard to determine the strong

*geodetic*number of a general

*graph*. ... Introduction Given two vertices u and v of a

*graph*, a geodesic is a

*shortest*

*path*between u and v. ...

##
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Page 3565 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

Summary: “The distance d(u,v) between two vertices u and v

*in*a connected*graph*G is the length of a*shortest*« —v*path**in*G. A u—v geodesic is a u—v*path*of length d(u,v). ... A vertex x lies*in*a u—v*path*P if x € V(P) — {u,v}. ...##
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On graphs with unique geoodesics and antipodes
[article]

2021
*
arXiv
*
pre-print

*In*1962, Oystein Ore asked

*in*which

*graphs*there is exactly one geodesic between any two vertices. He called such

*graphs*

*geodetic*. ...

*In*this paper, we systematically study properties of

*geodetic*

*graphs*, and also consider antipodal

*graphs*,

*in*which each vertex has exactly one antipode (a farthest vertex). ... Try to characterize these

*geodetic*

*graphs*

*in*other way." Definition 1. A geodesic between two vertices of a

*graph*is a

*shortest*

*path*connecting these vertices. Definition 2. ...

##
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Convex cycle bases

2013
*
Ars Mathematica Contemporanea
*

Convex cycles play a role e.g.

doi:10.26493/1855-3974.226.0a2
fatcat:2ftt4oixtfcsrfkcema5p6osfy
*in*the context of product*graphs*. ... We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given*graph*has a convex cycle basis and provides an explicit construction*in*the positive case. ... The Austrian participation*in*GReGAS is not supported by the Austrian Science Fonddation (FWF). ...##
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Preliminary studyon electronic map based on surface of ellipsoid

2009
*
Procedia Earth and Planetary Science
*

The paper investigates the fundamental method of

doi:10.1016/j.proeps.2009.09.178
fatcat:3v3ezwbdlrh7rbeljsvrlfalpe
*drawing*, display and analysis of the digital map on ellipsoid surface farther. ... Based on analysis of the limitations lying*in*the flat map, this paper introduces the concept of the digital map and electronic map on ellipsoid surface, elaborates on the superiority. ... We mainly study how to*draw*up the map*graph*on ellipsoid surface when we*draw*up electronic map on ellipsoid surface. ...##
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On networks with order close to the Moore bound
[article]

2021
*
arXiv
*
pre-print

*In*this paper we present new bounds on the order of mixed

*graphs*with given diameter or

*geodetic*girth and exhibit new examples of directed and mixed

*geodetic*cages. ...

*In*particular, we show that any k-

*geodetic*mixed

*graph*with excess one must have

*geodetic*girth two and be totally regular, thereby proving an earlier conjecture of the authors. ... Acknowledgements The first author acknowledges funding from an LMS Early Career Fellowship and thanks the Open University for an extension of funding

*in*2020. ...

##
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On total regularity of mixed graphs with order close to the Moore bound
[article]

2018
*
arXiv
*
pre-print

It is also of interest to find smallest possible k-

arXiv:1811.00650v1
fatcat:xvtielh6bjbtja5snncuewb7si
*geodetic*mixed*graphs*with minimum undirected degree ≥ r and minimum directed out-degree ≥ z. ... We finally put these results to practical use by proving the uniqueness of a 2-*geodetic*mixed*graph*with order exceeding the Moore bound by one. ... Conditions on r and z for the existence of such*graphs*for k = 2 are given*in*[25] and small k-*geodetic*mixed*graphs*are presented*in*[24] , along with lower bounds on the order of totally regular k-*geodetic*...##
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Mutual Visibility in Graphs
[article]

2021
*
arXiv
*
pre-print

Let G=(V,E) be a

arXiv:2105.02722v2
fatcat:xd6heouek5babic4pwgctrym2m
*graph*and P⊆ V a set of points. Two points are mutually visible if there is a*shortest**path*between them without further points. ...*In*this paper we start the study about this new invariant and the mutual-visibility sets*in*undirected*graphs*. ... This result can be easily generalized to*geodetic**graphs*: a*graph*is*geodetic*if the*shortest**path*between any pair of vertices is unique, like*in*block*graphs*and trees. ...##
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Further Towards Unambiguous Edge Bundling: Investigating Power-Confluent Drawings for Network Visualization
[article]

2019
*
arXiv
*
pre-print

Bach et al. [1] recently presented an algorithm for constructing confluent

arXiv:1810.09948v4
fatcat:izs3ngkgy5gjdjcklk6mfja3ga
*drawings*, by leveraging power*graph*decomposition to generate an auxiliary routing*graph*. ... We also classify the exact type of confluent*drawings*that the algorithm can produce as 'power-confluent', and prove that it is a subclass of the previously studied 'strict confluent'*drawing*. ... To*draw*the adjacency edges back on top, we can now forgo any*shortest**path*calculations. ...
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