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### Drawing Shortest Paths in Geodetic Graphs [article]

Sabine Cornelsen, Maximilian Pfister, Henry Förster, Martin Gronemann, Michael Hoffmann, Stephen Kobourov, Thomas Schneck
2020 arXiv   pre-print
Motivated by the fact that 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.  ...

### Hardness and approximation for the geodetic set problem in some graph classes [article]

Dibyayan Chakraborty, Florent Foucaud, Harmender Gahlawat, Subir Kumar Ghosh, Bodhayan Roy
2019 arXiv   pre-print
A set of vertices S of a 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  ...

### Strong geodetic problems in networks

Andrew Arockiaraj, Sandi Klavžar, Paul Manuel, Elizabeth Thomas, Antony Xavier
2018 Discussiones Mathematicae Graph Theory
The strong 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.  ...

### Algorithms and complexity for geodetic sets on planar and chordal graphs [article]

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, Bodhayan Roy
2020 arXiv   pre-print
A set S of vertices of a 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.  ...

### Strong geodetic problem in networks: computational complexity and solution for Apollonian networks [article]

Paul Manuel, Sandi Klavžar, Antony Xavier, Andrew Arokiaraj, Elizabeth Thomas
2017 arXiv   pre-print
The 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.  ...

### 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.  ...

### An O(mn^2) algorithm for computing the strong geodetic number in outerplanar graphs

Mauro Mezzini
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.  ...

### 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}.  ...

### On graphs with unique geoodesics and antipodes [article]

Dmitriy Gorovoy, David Zmiaikou
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.  ...

### Convex cycle bases

Marc Hellmuth, Josef Leydold, Peter F. Stadler
2013 Ars Mathematica Contemporanea
Convex cycles play a role e.g. 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).  ...

### Preliminary studyon electronic map based on surface of ellipsoid

Yong-chong Yang, You-yi Jian
2009 Procedia Earth and Planetary Science
The paper investigates the fundamental method of 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.  ...

### On networks with order close to the Moore bound [article]

James Tuite, Grahame Erskine
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.  ...

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

James Tuite, Grahame Erskine
2018 arXiv   pre-print
It is also of interest to find smallest possible k-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  ...

### Mutual Visibility in Graphs [article]

Gabriele Di Stefano
2021 arXiv   pre-print
Let G=(V,E) be a 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.  ...

### Further Towards Unambiguous Edge Bundling: Investigating Power-Confluent Drawings for Network Visualization [article]

Jonathan X. Zheng, Samraat Pawar, Dan F. M. Goodman
2019 arXiv   pre-print
Bach et al. [1] recently presented an algorithm for constructing confluent 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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