| Summary: | Cicerone and Di Stefano defined and studied the class of <i>k</i>-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most <i>k</i> times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the <i>k</i>-distance-hereditary graphs such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.
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