Abstract
A homogeneous set is a non-trivial module of a graph, i.e. a non-empty, non-unitary, proper subset of a graph's vertices such that all its elements present exactly the same outer neighborhood. Given two graphs \(G_1(V,E_1),\) \(G_2(V,E_2),\) the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a sandwich graph \(G_S(V,E_S)$, $E_1 \subseteq E_S \subseteq E_2,\) which has a homogeneous set. In 2001 Tang et al. published an all-fast \(O(n^2 \triangle_2)\) algorithm which was recently proven wrong, so that the HSSP's known upper bound would have been reset thereafter at the former \(O(n^4)\) determined by Cerioli et al. in 1998. We present, notwithstanding, new deterministic algorithms which have it established at \(O(n^3 \log ({m}/{n})).\) We give as well two even faster \(O(n^3)\) randomized algorithms, whose simplicity might lend them didactic usefulness. We believe that, besides providing efficient easy-to-implement procedures to solve it, the study of these new approaches allows a fairly thorough understanding of the problem.
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de Figueiredo, C., da Fonseca, G., de Sa, V. et al. Algorithms for the Homogeneous Set Sandwich Problem. Algorithmica 46, 149–180 (2006). https://doi.org/10.1007/s00453-005-1198-2
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DOI: https://doi.org/10.1007/s00453-005-1198-2