Bandwidth and density for block graphs
  The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete. 
Le Tu Quoc Hung and Maciej M. Syslo (University of Wroclaw), Margaret   L. Weaver (Eastern Illinois University), and Douglas...