On pitfalls in computing the geodetic number of a graph

Hansen, Pierre; Omme, Nikolaj van

doi:10.1007/s11590-006-0032-3

Cited by 15 publications

(13 citation statements)

References 8 publications

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“…The geodetic number g(G) of G is the size of a smallest set S ⊆ V (G) such that ∪ {u,v}∈( S 2 ) I(u, v) = V (G). The classical geodetic problem for the graph G is to determine g(G), see [5,7,9,12,23,24] for related investigations and in particular [4,17] for the geodetic number of Cartesian products. Very recently it was proved in [6] that deciding whether the geodetic number of a graph is at most k is NP-complete for graphs of maximum degree three.…”

Section: Preliminariesmentioning

confidence: 99%

Strong geodetic cores and Cartesian product graphs

Gledel

Iršič

Klavžar

2019

Applied Mathematics and Computation

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The strong geodetic problem on a graph G is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of G are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that sg(G K 2 ) ≥ sg(G) holds for different families of graphs, a result conjectured to be true in general. Counterexamples are constructed demonstrating that the conjecture does not hold in general.

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Section: Preliminariesmentioning

confidence: 99%

Strong geodetic cores and Cartesian product graphs

Gledel

Iršič

Klavžar

2019

Applied Mathematics and Computation

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“…Further results involving computational complexity problems related to geodesic convexity in graphs can be found in [3,4,6,38,80,81,83,84,86,95,104,110,113,123,140,164].…”

Section: Theorem 74 ([104]) the Convexity Number Problem Is Np-compmentioning

confidence: 99%

Geodesic Convexity in Graphs

Pelayo¹

2013

SpringerBriefs in Mathematics

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SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied School of Agricultural Engineering of Barcelona

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“…Geodetic number is the minimal-cardinality set of nodes, such that all shortest paths between its elements cover every node of the graph [16]. Calculating the geodetic number proved to be an NP-hard problem for general graphs [5].…”

Section: Introductionmentioning

confidence: 99%

“…Calculating the geodetic number proved to be an NP-hard problem for general graphs [5]. The integer linear programming (ILP) formulation of geodetic number problem was given in [16], containing also the first computational experiments on a set of random graphs.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Regression for Approximating Graph Geodetic Number

2021

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Graph properties are certain attributes that could make the structure of the graph understandable. Occasionally, standard methods cannot work properly for calculating exact values of graph properties due to their huge computational complexity, especially for real-world graphs. In contrast, heuristics and metaheuristics are alternatives proved their ability to provide sufficient solutions in a reasonable time. Although in some cases, even heuristics are not efficient enough, where they need some not easily obtainable global information of the graph. The problem thus should be dealt in completely different way by trying to find features that related to the property and based on these data build a formula which can approximate the graph property. In this work, symbolic regression with an evolutionary algorithm called Cartesian Genetic Programming has been used to derive formulas capable to approximate the graph geodetic number which measures the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Finding the exact value of the geodetic number is known to be NP-hard for general graphs. The obtained formulas are tested on random and real-world graphs. It is demonstrated how various graph properties as training data can lead to diverse formulas with different accuracy. It is also investigated which training data are really related to each property.

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On pitfalls in computing the geodetic number of a graph

Cited by 15 publications

References 8 publications

Strong geodetic cores and Cartesian product graphs

Strong geodetic cores and Cartesian product graphs

Geodesic Convexity in Graphs

Symbolic Regression for Approximating Graph Geodetic Number

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