Abstract
A median (antimedian) of a profile of vertices on a graph G is a vertex that minimizes (maximizes) the remoteness value, that is, the sum of the distances to the elements in the profile. The median (or antimedian) function has as output the set of medians (antimedians) of a profile. It is one of the basic models for the location of a desirable (or obnoxious) facility in a network. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper an axiomatic characterization is obtained for the median and antimedian functions on cocktail-party graphs. In addition a characterization of the antimedian function on complete graphs is presented.
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References
Arrow, K.: Social Choice and Individual Values, 1st edn. Cowles Commission for Research in Economics - Monographs, vol. 12. Wiley, New York (1951)
Arrow, K.J., Sen, A.K., Suzumura, K. (eds.): Handbook of Social Choice and Welfare, vol. 1. North Holland, Amsterdam (2002)
Arrow, K.J., Sen, A.K., Suzumura, K. (eds.): Handbook of Social Choice and Welfare, vol. 1. North Holland, Amsterdam (2005)
Balakrishnan, K., Brešar, B., Changat, M., Klavžar, S., Imrich, W., Kovše, M., Subhamathi, A.R.: On the Remoteness Function in Median Graphs. Discrete Appl. Math. 157, 3679–3688 (2009)
Balakrishnan, K., Brešar, B., Changat, M., Klavžar, S., Kovše, M., Subhamathi, A.R.: Computing Median and Antimedian Sets in Median Graphs. Algorithmica 57, 207–216 (2010)
Balakrishnan, K., Brešar, B., Changat, M., Klavžar, S., Kovše, M., Subhamathi, A.R.: Simultaneous Embedding of Graphs as Median and Antimedian Subgraphs. Networks 56, 90–94 (2010)
Balakrishnan, K., Changat, M., Klavžar, S., Joseph, M., Peterin, I., Prasanth, G.N., Špacapan, S.: Antimedian Graphs. Australas. J. Combin. 41, 159–170 (2008)
Balakrishnan, K., Changat, M., Mulder, H.M., Subhamathi, A.R.: Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes. Discrete Math. Algorithm. Appl. 04, 1250054, 20 pages (2012)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Heidelberg (1997)
Holzman, R.: An Axiomatic Approach to Location on Networks. Math. Oper. Res. 15, 553–563 (1990)
Klavžar, S., Mulder, H.M.: Median Graphs- Characterizations, Location Theory and Related Structures. J. Combin. Math. Combin. Comput. 30, 103–127 (1999)
McMorris, F.R., Mulder, H.M., Novick, B., Powers, R.C.: Five Axioms for Location Functions on Median Graphs. To appear in Discrete Math. Algorithms Appl.
McMorris, F.R., Mulder, H.M., Novick, B., Powers, R.C., Vohra, R.V.: Axiomatic characterization of voting procedures on K n (manuscript submitted)
McMorris, F.R., Mulder, H.M., Ortega, O.: Axiomatic Characterization of the Mean Function on Trees. Discrete Math. Algorithms Applications 2, 313–329 (2010)
McMorris, F.R., Mulder, H.M., Ortega, O.: Axiomatic Characterization of the ℓ p -Function on Trees. Networks 60, 94–102 (2012)
McMorris, F.R., Mulder, H.M., Roberts, F.S.: The Median Procedure on Median Graphs. Discrete Appl. Math. 84, 165–181 (1998)
McMorris, F.R., Mulder, H.M., Vohra, R.V.: Axiomatic Characterization of Location Functions. In: Kaul, H., Mulder, H.M. (eds.) Advances in Interdisciplinary Applied Discrete Mathematics, Interdisciplinary Mathematical Sciences, vol. 11, pp. 71–91. World Scientific Publishing, Singapore (2010)
McMorris, F.R., Roberts, F.S., Wang, C.: The Center Function on Trees. Networks 38, 84–87 (2001)
Minieka, E.: Anticenters and Antimedians of a Network. Networks 13, 35–364 (1983)
Mulder, H.M.: The Interval Function of a Graph. Math. Centre Tracts, vol. 132. Math. Centre, Amsterdam (1980)
Mulder, H.M.: Median Graphs. A Structure Theory. In: Kaul, H., Mulder, H.M. (eds.) Advances in Interdisciplinary Applied Discrete Mathematics. Interdisciplinary Mathematical Sciences, vol. 11, pp. 93–125. World Scientific Publishing, Singapore (2010)
Mulder, H.M., Novick, B.A.: An Axiomization of the Median Function on the n-Cube. Discrete Appl. Math. 159, 139–144 (2011)
Mulder, H.M., Novick, B.A.: A Tight Axiomatization of the Median Function on Median Graphs. Discrete Appl. Math. 161, 838–846 (2013)
Mulder, H.M., Reid, K.B., Pelsmajer, M.J.: Axiomatization of the Center Function on Trees. Australasian J. Combin. 41, 223–226 (2008)
Rao, S.B., Vijayakumar, A.: On the Median and the Antimedian of a Cograph. Int. J. Pure Appl. Math. 46, 703–710 (2008)
Vohra, R.: An Axiomatic Characterization of Some Locations in Trees. European J. Operational Research 90, 78–84 (1996)
Shilpa, M., Changat, M., Narasimha-Shenoi, P.G.: Axiomizatic Characterization of the Center Function on Some Graph Classes (manuscript submitted)
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Changat, M., Lekha, D.S., Mulder, H.M., Subhamathi, A.R. (2015). Axiomatic Characterization of the Median and Antimedian Functions on Cocktail-Party Graphs and Complete Graphs. In: Ganguly, S., Krishnamurti, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2015. Lecture Notes in Computer Science, vol 8959. Springer, Cham. https://doi.org/10.1007/978-3-319-14974-5_14
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DOI: https://doi.org/10.1007/978-3-319-14974-5_14
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