Abstract
Let G = ( V , E ) be a connected graph. The distance between two vertices x and y in G , denoted by d G ( x , y ) , is the length of a shortest path between x and y . A graph G is called almost distance-hereditary, if each connected induced subgraph H of G has the property that d H ( u , v ) ≤ d G ( u , v ) + 1 for every pair of vertices u and v in H . We will confirm that every 2-connected, claw-free and almost distance-hereditary graph has a Hamiltonian cycle.
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