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On pitfalls in computing the geodetic number of a graph

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Abstract

Given a simple connected graph G = (V, E) the geodetic closure \(I[S] \subset V\) of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes \(v_k,v_l \in S\). The geodetic number, denoted by g(G), is the smallest cardinality of a node set S * such that I[S *] =  V. In “The geodetic number of a graph”, [Harary et al. in Math. Comput. Model. 17:89–95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0–1 integer programming model to find the geodetic number. Computational results are given.

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Authors and Affiliations

  1. GERAD and HEC Montréal, Montreal, Canada

    Pierre Hansen

  2. CRT and École Polytechnique de Montréal, Montreal, Canada

    Nikolaj van Omme

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  1. Pierre Hansen
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  2. Nikolaj van Omme
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Correspondence to Pierre Hansen.

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Hansen, P., van Omme, N. On pitfalls in computing the geodetic number of a graph. Optimization Letters 1, 299–307 (2007). https://doi.org/10.1007/s11590-006-0032-3

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  • DOI: https://doi.org/10.1007/s11590-006-0032-3

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