Geodesic
Classes of graphs with restricted interval models
Discrete mathematics & theoretical computer science, Tome 3 (1998-1999) no. 4.

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We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment. 
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     author = {Proskurowski, Andrzej and Telle, Jan Arne},
     title = {Classes of graphs with restricted interval models},
     journal = {Discrete mathematics & theoretical computer science},
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Proskurowski, Andrzej; Telle, Jan Arne. Classes of graphs with restricted interval models. Discrete mathematics & theoretical computer science, Tome 3 (1998-1999) no. 4. doi : 10.46298/dmtcs.263. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.263/

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