Abstract
Let G be a connected graph and S ⊆ V( G). Then the Steiner distance of S in G, denoted by d G ( S), is the smallest number of edges in a connected subgraph of G that contains S. A connected graph is k-Steiner distance hereditary, k ⩾ 2, if for every S ⊆ V( G) such that ¦ S ¦ = k and every connected induced subgraph H of G containing S, d H ( S) = d G ( S). A polynomial algorithm for testing whether a graph is 3-Steiner distance hereditary is developed. In addition, a polynomial algorithm for testing whether an arbitrary graph has a cycle of length exceeding t, for any fixed t, without crossing chords is provided.
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