Abstract
Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that \(ecc_T(v)-ecc_G(v)\le 2\) for every vertex v, where \(ecc_G(v)\) (\(ecc_T(v)\)) is the eccentricity of a vertex v in G (in T, respectively). Using only metric properties of graphs, we extend that result to a much larger family of graphs containing among others chordal graphs and the underlying graphs of 7-systolic complexes. Furthermore, based on our approach, we propose two heuristics for constructing eccentricity k-approximating trees with small values of k for general unweighted graphs. We validate those heuristics on a set of real-world networks and demonstrate that all those networks have very good eccentricity approximating trees.
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References
Bandelt, H.-J., Chepoi, V.: 1-hyperbolic graphs. SIAM J. Discret. Math. 16, 323–334 (2003)
Brandstädt, A., Chepoi, V., Dragan, F.F.: Distance approximating trees for chordal and dually chordal graphs. J. Algorithms 30, 166–184 (1999)
Chepoi, V.: Some \(d\)-convexity properties in triangulated graphs. Math. Res. 87, 164–177, Ştiinţa, Chişinău (1986). (Russian)
Chepoi, V.: Centers of triangulated graphs. Math. Notes 43, 143–151 (1988)
Chepoi, V.: Graphs of some CAT(0) complexes. Adv. Appl. Math. 24, 125–179 (2000)
Dragan, F.F.: Centers of graphs and the Helly property (in Russian). Ph.D. thesis, Moldova State University (1989)
Farber, M., Jamison, R.E.: On local convexity in graphs. Discret. Math. 66, 231–247 (1987)
Kratsch, D., Le, H.-O., Müller, H., Prisner, E., Wagner, D.: Additive tree spanners. SIAM J. Discret. Math. 17, 332–340 (2003)
Madanlal, M.S., Vankatesan, G., Pandu Rangan, C.: Tree 3-spanners on interval, permutation and regularbipartite graphs. Inf. Process. Lett. 59, 97–102 (1996)
Nandakumar, R., Parthasarathy, K.: Eccentricity-preserving spanning trees. J. Math. Phys. Sci. 24, 33–36 (1990)
Prisner, E.: Distance approximating spanning trees. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 499–510. Springer, Heidelberg (1997)
Prisner, E.: Eccentricity-approximating trees in chordal graphs. Discret. Math. 220, 263–269 (2000)
Soltan, V.P., Chepoi, V.D.: Conditions for invariance of set diameters under \(d\)-convexification in a graph. Cybernetics 19, 750–756 (1983). (Russian, English transl.)
Yushmanov, S.V., Chepoi, V.: A general method of investigation of metric graphproperties related to the eccentricity. In: Mathematical Problems in Cybernetics, vol. 3, pp. 217–232. Nauka, Moscow (1991). (Russian)
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Dragan, F.F., Köhler, E., Alrasheed, H. (2016). Eccentricity Approximating Trees. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_13
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DOI: https://doi.org/10.1007/978-3-662-53536-3_13
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