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

Dibyayan Chakraborty, Florent Foucaud, Harmender Gahlawat, Subir Kumar Ghosh, Bodhayan Roy
<span title="2019-09-19">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we prove that solving the MGS problem is NP-hard on planar graphs with a maximum degree six and line graphs.  ...  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.  ...  P d ) contains C V I T 2 0 1 6 23:10 Hardness and approximation for the geodetic set problem in some graph classes se(X) (resp. ne(X)).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.08795v1">arXiv:1909.08795v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fpe4dfo6l5f4xltxjwescdepk4">fatcat:fpe4dfo6l5f4xltxjwescdepk4</a> </span>
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### Hull and Geodetic Numbers for Some Classes of Oriented Graphs [article]

Julio C. S. Araujo, Pedro S. M. Arraes
<span title="2020-11-23">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
S ⊆ V(D) is a geodetic set of D if each vertex of D lies in a (u,v)-geodesic, for some u,v ∈ S.  ...  S ⊆ V(D) is convex if, for every u,v ∈ S, the vertices in each (u,v)-geodesic and in each (v,u)-geodesic are in S.  ...  Another natural problem is to find some graph class G for which determining − → hn(D) is a NP-hard problem, while determining hn(G) can be solved in polynomial time, for some simple graph G ∈ G and some  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.10240v2">arXiv:1911.10240v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dsq5tjglgre7loinlqwh42xgqq">fatcat:dsq5tjglgre7loinlqwh42xgqq</a> </span>
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### On the geodetic rank of a graph

Mamadou Moustapha Kanté, Rudini M. Sampaio, Vinícius F. dos Santos, Jayme L. Szwarcfiter
<span title="">2017</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ppxcitfjyjcpjl6wlzzfryrguu" style="color: black;">Journal of Combinatorics</a> </i> &nbsp;
Among the results, we prove that it is NP-hard to approximate the geodetic rank of bipartite graphs by a factor of n 1−ε , for every ε > 0.  ...  The graph convexity is geodetic when its convex sets are closed under shortest paths. For a subset S ⊆ V (G), the smallest convex set containing S, denoted by H(S), is the hull of S.  ...  We would like to thank Manoel Campêlo, UFC, Fortaleza, Brazil, for his valuable comments.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/joc.2017.v8.n2.a5">doi:10.4310/joc.2017.v8.n2.a5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ljrfxl46srcbbpqui5m3vcyxca">fatcat:ljrfxl46srcbbpqui5m3vcyxca</a> </span>
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### Preface: LAGOS'13: Seventh Latin-American Algorithms, Graphs, and Optimization Symposium, Playa del Carmen, México — 2013

José Correa, Guillermo Durán, Luerbio Faria, Miguel Pizaña, Gelasio Salazar
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Dourado, Lucia Penso, and Dieter Rautenbach, shows that it is NP-hard to minimize losses in burning trees in a weighted version of Hartnell's firefighter game (1995), but also provides two approximation  ...  A survey-based approach for designing the lines of a rapid transit network, by L.F. Escudero and S.  ...  and the Sociedad Matemática Mexicana for their financial support, which made the organization of this event possible.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.05.007">doi:10.1016/j.dam.2016.05.007</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tbv2l3vfjnhenltu7ipqrn23hu">fatcat:tbv2l3vfjnhenltu7ipqrn23hu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170923054842/http://repositorio.uchile.cl/bitstream/handle/2250/141842/Preface-LAGOS-13.pdf?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/76/6a/766aaee1277b41d5f9a07e10b06783c0e95a36ea.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.05.007"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### On pitfalls in computing the geodetic number of a graph

Pierre Hansen, Nikolaj van Omme
<span title="2006-11-08">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5dkx5l3pkjclfd46nyh4eftsrq" style="color: black;">Optimization Letters</a> </i> &nbsp;
In "The geodetic number of a graph", Mathematical and Computer Modelling 17 (June 1993) 89-95, F. Harary, E. Loukakis and C.  ...  Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ 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  ...  One can see that augmenting the number of nodes by 5 multiplies approximatively the computing time by 5. This illustrates the fact that the problem of finding the geodetic number is NP-Hard.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11590-006-0032-3">doi:10.1007/s11590-006-0032-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o72z6hc3xneffn5hreviqvabou">fatcat:o72z6hc3xneffn5hreviqvabou</a> </span>
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### Further hardness results on rainbow and strong rainbow connectivity

Juho Lauri
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
We conclude by considering some tractable special cases, and show for instance that both problems are in when parameterized by tree-depth.  ...  Previously, no graph class was known where the complexity of the two problems would differ.  ...  Acknowledgements The author thanks Henri Hansen, Mikko Lauri, and Keijo Ruohonen for helpful comments, and acknowledges the idea of Henri Hansen that led to Theorem 9.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2015.07.041">doi:10.1016/j.dam.2015.07.041</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/brax3buo7jbwzgfvwf2luyrqqy">fatcat:brax3buo7jbwzgfvwf2luyrqqy</a> </span>
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### Computing Minimum Geodetic Sets of Proper Interval Graphs [chapter]

Tınaz Ekim, Aysel Erey, Pinar Heggernes, Pim van 't Hof, Daniel Meister
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We show that the geodetic number of proper interval graphs can be computed in polynomial time. This problem is NP-hard on chordal graphs and on bipartite weakly chordal graphs.  ...  Only an upper bound on the geodetic number of proper interval graphs has been known prior to our result.  ...  algorithm to approximate the geodetic number of bipartite permutation graphs with an additive factor 1, and a proof of NP-hardness of computing the geodetic number of cobipartite graphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-29344-3_24">doi:10.1007/978-3-642-29344-3_24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/whktyut3yvge7dsjkqosjffan4">fatcat:whktyut3yvge7dsjkqosjffan4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190303142741/http://pdfs.semanticscholar.org/caff/871ef162f059c1568202d0eeef3050454913.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ca/ff/caff871ef162f059c1568202d0eeef3050454913.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-29344-3_24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

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

Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, Bodhayan Roy
<span title="2020-06-30">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
(CALDAM, 2020) and show that it remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature.  ...  The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph.  ...  We have seen that for every fixed k, MGS is polynomial-time for k-trees, and this is unlikely to be the case for partial k-trees by [18] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.16511v1">arXiv:2006.16511v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ax56pso6ynb4bmc3jb2yf6rt6i">fatcat:ax56pso6ynb4bmc3jb2yf6rt6i</a> </span>
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### Geodetic Topological Cycles in Locally Finite Graphs

Agelos Georgakopoulos, Philipp Sprüssel
<span title="2009-11-30">2009</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We prove that the topological cycle space ${\cal C}(G)$ of a locally finite graph $G$ is generated by its geodetic topological circles.  ...  We further show that, although the finite cycles of $G$ generate ${\cal C}(G)$, its finite geodetic cycles need not generate ${\cal C}(G)$.  ...  Acknowledgement The counterexamples in Figures 3.1 and 5.5 are due to Henning Bruhn. The authors are indebted to him for these counterexamples and other useful ideas.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/233">doi:10.37236/233</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ofp2dnzemzcutfrjjikxpdpute">fatcat:ofp2dnzemzcutfrjjikxpdpute</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809100004/http://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r144/pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/64/29/64294b6b015d02e752c9dfda304f75b4f7a1e2f9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/233"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### Geodetic topological cycles in locally finite graphs [article]

Agelos Georgakopoulos, Philipp Sprüssel
<span title="2009-11-20">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles.  ...  We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G).  ...  Acknowledgement The counterexamples in Figures 3.1 and 5.5 are due to Henning Bruhn. The authors are indebted to him for these counterexamples and other useful ideas.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0911.3999v1">arXiv:0911.3999v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7bv7bmuduremzl43lxzg4di4ii">fatcat:7bv7bmuduremzl43lxzg4di4ii</a> </span>
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### A Method for Estimating Surveillance Video Georeferences

<span title="2017-07-09">2017</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fw2uac7a5jggbfgqwh2wqy7yxu" style="color: black;">ISPRS International Journal of Geo-Information</a> </i> &nbsp;
A camera view in 3D space is completely determined by the camera position, orientation, and field-of-view.  ...  Since the accurate measuring of these parameters can be extremely difficult, in this paper we propose a method for their estimation based on matching video frame coordinates of certain point features with  ...  The estimation error for a camera's field-of-view is depicted in the next graph ( Figure 9 ).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/ijgi6070211">doi:10.3390/ijgi6070211</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yrrkmsdqqnad7oru2c6or244xm">fatcat:yrrkmsdqqnad7oru2c6or244xm</a> </span>
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### Tomographic Node Placement Strategies and the Impact of the Routing Model

Yvonne-Anne Pignolet, Stefan Schmid, Gilles Tredan
<span title="2018-06-12">2018</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pkal2zplnrarzkkzugv4u5ithy" style="color: black;">Performance Evaluation Review</a> </i> &nbsp;
In particular, we provide optimal and near-optimal algorithms to deploy a minimal number of asymmetric and symmetric tomography nodes for basic network topologies (modelled as graphs) under different routing  ...  This facilitates algorithmic reasoning about tomographic placement problems and quantifying the difference between routing models.  ...  We thank Michael Markovitch for many discussions in the early stage of this work. We are also grateful for the feedback and support from our SIGMETRICS shepherd Paul Barford.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3292040.3219648">doi:10.1145/3292040.3219648</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ay2evmpmfbfencpzsq4jumxd2m">fatcat:ay2evmpmfbfencpzsq4jumxd2m</a> </span>
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### Tomographic Node Placement Strategies and the Impact of the Routing Model

Yvonne-Anne Pignolet, Stefan Schmid, Gilles Tredan
<span title="2019-01-17">2019</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pkal2zplnrarzkkzugv4u5ithy" style="color: black;">Performance Evaluation Review</a> </i> &nbsp;
In particular, we provide optimal and near-optimal algorithms to deploy a minimal number of asymmetric and symmetric tomography nodes for basic network topologies (modelled as graphs) under different routing  ...  This facilitates algorithmic reasoning about tomographic placement problems and quantifying the difference between routing models.  ...  We thank Michael Markovitch for many discussions in the early stage of this work. We are also grateful for the feedback and support from our SIGMETRICS shepherd Paul Barford.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3308809.3308849">doi:10.1145/3308809.3308849</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xipeflguunchzm63d7rwuh3efe">fatcat:xipeflguunchzm63d7rwuh3efe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190502111345/https://hal.archives-ouvertes.fr/hal-01644582/file/sigmetrics18path-hal.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6d/43/6d430a88a28f79b3b0459d9ccae3811176cab339.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3308809.3308849"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

### Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs [article]

Melissa Keranen, Juho Lauri
<span title="2018-05-25">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time.  ...  Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an n-vertex split graph with a factor of n^1/2-ϵ for any ϵ > 0 unless P = NP.  ...  For every k ≥ 3, the k-SSRC problem is NP-complete when the graph G is a star. We reduce from this problem, and make use of some ideas of [4] in the following.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1405.6893v7">arXiv:1405.6893v7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gobg6k74yreybbn5s3hruy6gbm">fatcat:gobg6k74yreybbn5s3hruy6gbm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826095259/https://arxiv.org/pdf/1405.6893v7.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/7c/61/7c613d9b3579f9c1ff3ce3ab4a886fd533bd4113.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1405.6893v7" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Complexity of rainbow vertex connectivity problems for restricted graph classes

Juho Lauri
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Moreover, we show both problems are solvable in polynomial time for block graphs, while is tractable for cactus graphs and split graphs.  ...  We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k-regular graphs for k ≥ 3.  ...  Hardness results In this section, we will give a number of hardness results for both Rainbow Vertex Connectivity and Strong Rainbow Vertex Connectivity for very restricted graph classes.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.11.023">doi:10.1016/j.dam.2016.11.023</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h5ud5mtswzfizeg5jvdbd6yupy">fatcat:h5ud5mtswzfizeg5jvdbd6yupy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200916061625/https://arxiv.org/pdf/1612.07768v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9c/5d/9c5dfe730a3a2427adf8de6c16f35b55a3d77c0e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2016.11.023"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>
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