Some theorems on graphs and posets

Lcclerc proved that the dimension of the partiailly ordered slet consisting of all subf~ce'~ of a tree T, m&red by inclusion, is the number of end yuints of 'I'. Leclerc posed the probkrn of determitAng the dimension the partially ed set P consisting of all inducxxI connected subgraphs of a connec graph G for tPisa lattice. In this paper, WC prove that the poset P consisting af all induced connected subgraphs of a nontrivial connected graph G, partially ordered by inclusion, has dimcnsior. n whcrc n is the number of rtioncut vertices in G wheehcr or not P is a fatticc. We also dctcrmine the ditnensian of the distrilbutivc Iattic?-of all subgraphs of a 6 {ph. The dimension of' a partialfy ordered set (X, P) was defi lm!l ] as H-w minimum nuflhcr of linear orders on intersection is l? A goset has ciiinensicm one if and only if it is a chain. In [ 61 Leclorc proved the fol2owing result which gives an alternate rmitim of dismzsion for hs~,: posets which are not chains.