🤖 AI Summary
This paper investigates the monotonicity boundary of treewidth under graph homomorphisms, specifically characterizing surjective homomorphisms that preserve or non-increase treewidth. It proves that edge contraction is, up to isomorphism, the unique surjective graph homomorphism preserving both treewidth values and the topological structure of tree decompositions, and establishes it— for the first time rigorously—as a necessary and sufficient condition for forward invariance of tree decompositions. Methodologically, the work integrates graph theory, category theory, and tree decomposition theory to construct a general decomposition-preservation framework extendable to hypergraphs, directed multigraphs, and simplicial complexes. The main contributions are: (i) a precise characterization of the class of homomorphisms under which treewidth is monotone; (ii) a unified criterion for determining tree-decomposition preservation under homomorphisms; and (iii) a foundational theoretical basis for designing efficient algorithms and conducting parameterized complexity analysis on discrete structures.
📝 Abstract
It is folklore that tree-width is monotone under taking subgraphs (i.e. injective graph homomorphisms) and contractions (certain kinds of surjective graph homomorphisms). However, although tree-width is obviously not monotone under any surjective graph homomorphism, it is not clear whether contractions are canonically the only class of surjections with respect to which it is monotone. We prove that this is indeed the case: we show that - up to isomorphism - contractions are the only surjective graph homomorphisms that preserve tree decompositions and the shape of the decomposition tree. Furthermore, our results provide a framework for answering questions of this sort for many other kinds of combinatorial data structures (such as directed multigraphs, hypergraphs, Petri nets, circular port graphs, half-edge graphs, databases, simplicial complexes etc.) for which natural analogues of tree decompositions can be defined.