Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

The thinness of a graph is width parameter that generalizes some properties interval graphs, which are exactly the graphs one. Graphs with at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time bounded thinness, given suitable representation graph. Proper defined analogously, generalizing proper and larger family known to polynomially solvable thinness. complexity recognizing 2-thin still open. In this work, we present characterizations as intersection rectangles plane, vertex paths on grid (VPG graphs), by forbidden ordered patterns. We also prove independent bigraphs, permutation Finally, take step towards placing its variations landscape parameters, upper bounding terms bandwidth.