h-graphs: A new representation for tree decompositions of graphs

• h-graphs, a new representation for tree decompositions of constraints graph is presented.
• h-graphs explicitly capture construction steps dependencies in a tree decomposition.
• An application to speed up computing feasibility ranges for constraint parameters is described.

In geometric constraint solving, 2D well constrained geometric problems can be abstracted as Laman graphs. If the graph is tree decomposable, the constraint-based geometric problem can be solved by a Decomposition–Recombination planner based solver. In general decomposition and recombination steps can be completed only when steps on which they are dependent have already been completed. This fact naturally defines a hierarchy in the decomposition–recombination steps that traditional tree decomposition representations do not capture explicitly.In this work we introduce h-graphs, a new representation for decompositions of tree decomposable Laman graphs, which captures dependence relations between different tree decomposition steps. We show how h-graphs help in efficiently computing parameter ranges for which solution instances to well constrained, tree decomposable geometric constraint problems with one degree of freedom can actually be constructed.