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Block decomposition approach to compute a minimum geodeticset∗∗∗

Published online by Cambridge University Press:  10 June 2014

Tınaz Ekim  and
Aysel Erey
Tınaz Ekim
Affiliation:
Boğaziçi University, Department of Industrial Engineering, 34342 Bebek, Istanbul, Turkey.. tinaz.ekim@boun.edu.tr
Aysel Erey
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada.; eaysel@mathstat.dal.ca
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Abstract

In this paper, we develop a divide-and-conquer approach, called block decomposition, tosolve the minimum geodetic set problem. This provides us with a unified approach for allgraphs admitting blocks for which the problem of finding a minimum geodetic set containinga given set of vertices (g-extension problem) can be efficiently solved. Ourmethod allows us to derive linear time algorithms for the minimum geodetic set problem in(a proper superclass of) block-cacti and monopolar chordal graphs. Also, we show that hullsets and geodetic sets of block-cacti are the same, and the minimum geodetic set problemis NP-hard in cobipartite graphs. We conclude by pointing out several interesting researchdirections.

Keywords

Convexitygeodetic sethull setgraph classes
Type
Research Article
Information
RAIRO - Operations Research , Volume 48 , Issue 4 , October 2014 , pp. 497 - 507
Copyright
© EDP Sciences, ROADEF, SMAI 2014

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References

Bertossi, A.A., Dominating sets for split and bipartite graphs. Inf. Proc. Lett. 19 (1984) 37–40. Google Scholar
Brešar, B. and Tepeh Horvat, A., On the geodetic number of median graphs. Discrete Math. 308 (2008) 4044–4051. Google Scholar
Brešar, B., Klavžar, S. and Tepeh Horvat, A., On the geodetic number and related metric sets in Cartesian product graphs. Discrete Math. 308 (2008) 5555–5561. Google Scholar
Buckley, F. and Harary, F., Geodetic games for graphs. Questiones Math. 8 (1986) 321–334. Google Scholar
Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M. and Puertas, M.L., On the geodetic and the hull numbers in strong product graphs. Comput. Math. Appl. 60 (2010) 3020–3031. Google Scholar
Cáceres, J., Marquez, A., Oellerman, O. R., Puertas, M. L., Rebuilding convex sets in graphs. Discrete Math. 297 (2005) 26–37. Google Scholar
Canoy, S.R., Cagaanan, G.B. and Gervacio, S.V., Convexity, Geodetic and Hull Numbers of the Join of Graphs. Utilitas Mathematica 71 (2006) 143–159. Google Scholar
Chae, G.-B., Palmer, E.M. and Siu, W.C., Geodetic number of random graphs of diameter 2. Aust. J. Combinatorics 26 (2002) 11–20. Google Scholar
Dourado, M.C., Gimbel, J.G., Kratochvíl, J., Protti, F. and Szwarcfiter, J.L., On the computation of the hull number of a graph. Discrete Math. 309 (2009) 5668–5674. Google Scholar
Dourado, M.C., Protti, F., Rautenbach, D. and Szwarcfiter, J.L., Some remarks on the geodetic number of a graph. Discrete Math. 310 (2010) 832–837. Google Scholar
Ekim, T., Erey, A., Heggernes, P., van’t Hof, P. and Meister, D., Computing Minimum Geodetic Sets of Proper Interval Graphs. Proceedings of LATIN 2012. Lect. Notes Comput. Sci. 7256 (2012) 279–290. Google Scholar
Ekim, T., Hell, P., Stacho, J. and de Werra, D., Polar chordal graphs. Discrete Appl. Math. 156 (2008) 2469–2479. Google Scholar
A. Erey, Convexity in graphs. Master’s Thesis, Boğaziçi University (2011).
Farber, M. and Jamison, R.E., Convexity in graphs and hypergraphs. SIAM J. Algebraic Discrete Meth. 7 (1986) 433–442. Google Scholar
Gerstel, O. and Zaks, S., A new characterization of tree medians with applications to distributed sorting. Networks 24 (1994) 23–29. Google Scholar
Hansberg, A. and Volkmann, L., On the geodetic and geodetic domination numbers of a graph. Discrete Math. 310 (2010) 2140–2146. Google Scholar
Harary, F., Loukakis, E. and Tsouros, C., The geodetic number of a graph. Math. Comput. Modell. 17 (1993) 89–95. Google Scholar
Haynes, T.W., Henning, M.A. and Tiller, C., Geodetic achievement and avoidance games for graphs. Quaestiones Math. 26 (2003) 389–397. Google Scholar
Hernando, C., Jiang, T., Mora, M., Pelayo, I.M. and Seara, C., On the Steiner, geodetic and hull numbers of graphs Discrete Math. 293 (2005) 139–154. Google Scholar
Howorka, E., A characterization of ptolemaic graphs. J. Graph Theory 5 (1981) 323–331. Google Scholar
Kang, A.N.C. and Ault, D.A., Some properties of a centroid of a free tree. Inf. Process. Lett. 4 (1975) 18–20. Google Scholar
Kanté, M.M. and Nourine, L., Polynomial Time Algorithms for Computing a Minimum Hull Set in Distance-Hereditary and Chordal Graphs. Proceedings of SOFSEM 2013. Lect. Notes Comput. Sci. 7741 (2013) 268–279. Google Scholar
Mitchell, S.L., Another characterization of the centroid of a tree. Discrete Math. 24 (1978) 277–280. Google Scholar
H.M. Mulder, The Interval Function of a Graph. Mathematish Centrum, Tract. 132, Amsterdam (1981).
Necásková, M., A note on the achievement geodetic games. Questiones Math. 12 (1988) 115–119. Google Scholar
Pandu Rangan, C., Parthasarathy, K.R. and Prakash, V., On the g-centroidal problem in special classes of perfect graphs. Ars Combinatoria 50 (1998) 267–278. Google Scholar
F.S. Roberts, Indifference graphs, in Proof techniques in graph theory, edited by F. Harary, Academic Press (1969) 139–146.
Veeraraghavan, P., An efficient g-centroid location algorithm for cographs. Inter. J. Math. Math. Sci. 9 (2005) 1405–1413. Google Scholar
P. Veeraraghavan, Application of g-convexity in mobile ad hoc networks, in 6th International Conference on Information Technology in Asia 2009, Kuching, Sarawak, Malaysia, vol. CITA 09 (2009) 33–38.
M.J.L. van de Vel, Theory of convex structures. North-Holland (1993).
Wang, F.H., Wang, Y.L. and Chang, J.M., The lower and upper forcing geodetic numbers of block-cactus graphs. Eur. J. Oper. Res. 175 (2006) 238–245. Google Scholar