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Abstract: | The median (antimedian) set of a profile π = (u1, . . . , uk) of vertices of a graphG is the set of vertices x that minimize (maximize) the remoteness i d(x,ui ). Two algorithms for median graphs G of complexity O(nidim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes |
Description: | Algorithmica (2010) 57: 207–216 DOI 10.1007/s00453-008-9200-4 |
URI: | http://dyuthi.cusat.ac.in/purl/4193 |
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Computing median and antimedian sets in median.pdf | (290.4Kb) |
Abstract: | A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x 2 V.G/ the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O.mlog n/ time whether G is a median graph with geodetic number 2 |
Description: | Discrete Applied Mathematics 157 (2009) 3679- 3688 |
URI: | http://dyuthi.cusat.ac.in/purl/4197 |
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On the remoteness function in median graphs.pdf | (609.5Kb) |
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