On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets |
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Authors: | D. S. Taletskii D. S. Malyshev |
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Affiliation: | 1.Lobachevsky Nizhny Novgorod State University,Nizhny Novgorod,Russia;2.National Research University Higher School of Economics,Nizhny Novgorod,Russia |
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Abstract: | Given n and d, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of n-vertex trees of vertex degree at most d.We show that the extremal tree is unique for all even n but uniqueness may fail for odd n; moreover, for d = 3 and every odd n ≥ 7, there are exactly ?(n ? 3)/4? + 1 extremal trees. In the paper, the problem of searching for extremal (n, d)-trees is also considered for the 2-caterpillars; i.e., the trees in which every vertex lies at distance at most 2 from some simple path. Given n and d ∈ {3, 4}, we completely reveal all extremal 2-caterpillars on n vertices each of which has degree at most d. |
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