Some theorems on graphs and posets |
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Authors: | William T. Trotter John I. Moore |
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Affiliation: | Department of Mathematics and Computer Science, University of South Carolina, Columbia, SC 29208, U.S.A. |
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Abstract: | In this journal, Leclerc proved that the dimension of the partially ordered set consisting of all subtrees of a tree T, ordered by inclusion, is the number of end points of T. Leclerc posed the problem of determining the dimension of the partially ordered set P consisting of all induced connected subgraphs of a connected graph G for which P is a lattice.In this paper, we prove that the poset P consisting of all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimension n where n is the number of noncut vertices in G whether or not P is a lattice. We also determine the dimension of the distributive lattice of all subgraphs of a graph. |
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