Planar Graphs and Partial k-Trees

It is well-known that many NP-hard problems can be solved efficiently on graphs of bounded treewidth. We begin by showing that Knuth’s results on nested satisfiability are easily derived from this fact since nested satisfiability graphs have treewidth at most three. Noting that nested satisfiability graphs have a particular form of planar drawing, we define a more general form of graph drawing while maintaining planarity and bounded treewidth. We name all graphs with such drawings partial Matryoshka graphs—a reference to the nesting Russian dolls—and demonstrate that they encompass other important graph classes of treewidth at most three, such as Halin and IO-graphs. Based on decompositions derived for partial Matryoshka graphs, we then proceed to explore edge-decompositions into graphs of bounded treewidth. We prove that any Hamiltonian planar graph on n vertices can be decomposed into a forest and a graph of O(logn) treewidth, and provide an efficient algorithm for constructing this decomposition. Similarly, we show that graphs of maximum degree three can be decomposed into a matching and an SP-graph (i.e. a graph of treewidth two). Furthermore, we prove that this manner of decomposition cannot be extended to graphs of higher degree by giving a method for constructing a graph of maximum degree four that cannot be decomposed into a matching and a graph of treewidth at most k, for any constant k. Lastly, we motivate such decompositions by producing an approximation algorithm for weighted vertex cover and weighted independent set on all graphs that can be decomposed into a forest and a graph of bounded treewidth.