Abstract
Let G be a cube-free median graph. It is proved that k / 2 ⩾ n - 1 ⩾ m / 2 n ⩾ s ⩾ r - 1 , where n, m, s, k, and r are the number of vertices, edges, squares, Θ -classes, and the number of edges in a smallest Θ -class of G, respectively. Moreover, the equalities characterize Cartesian products of two trees of the same order. The cube polynomial of cube-free median graphs is also considered and it is shown that planar cube-free median graphs can be recognized in linear time.
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