Contracting planar graphs to contractions of triangulations

For every graph $H$, there exists a polynomial-time algorithm deciding if a planar input graph $G$ can be contracted to~$H$. However, the degree of the polynomial depends on the size of $H$. In this paper, we identify a class of graphs $\cal C$ such that for every $H \in \cal C$, there exists an algorithm deciding in time $f(|V(H)|) \cdot |V(G)|^{\bigO{1}}$ whether a planar graph $G$ can be contracted to~$H$. (The function $f(\cdot)$ does not depend on $G$.) The class $\cal C$ is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph $H \in \cal C$ if and only if there exists a constant $c_H$ such that if the tree-width of a graph is at least $c_H$, it contains $H$ as a contraction. We also provide a characterization of $\cal C$ in terms of minimal forbidden contractions.

💡 Research Summary

The paper addresses the computational problem of deciding whether a given planar graph G can be contracted to a fixed target graph H. While it is known that for any fixed H there exists a polynomial‑time algorithm for this decision, the degree of the polynomial typically depends on the size of H, making the approach impractical for larger targets. The authors therefore identify a natural subclass 𝒞 of graphs for which the dependence on H can be isolated into a separate factor f(|V(H)|), yielding an algorithm whose running time is f(|V(H)|)·|V(G)|^{O(1)}.

The class 𝒞 is defined as the closure of planar triangulations under the contraction operation. A planar triangulation is a planar embedding in which every face (including the outer face) is a triangle. By taking all possible edge‑contractions of such triangulations, the authors obtain a rich family that includes many non‑triangulated planar graphs but retains strong structural properties inherited from the underlying triangulations.

The central technical contribution is a two‑fold characterization of 𝒞. First, the authors prove that for every H∈𝒞 there exists a constant c_H such that any planar graph whose tree‑width is at least c_H necessarily contains H as a contraction. This “large tree‑width ⇒ forced contraction” theorem mirrors classic results in graph minor theory (e.g., the Robertson‑Seymour Grid‑Minor Theorem) but is tailored to the contraction operation rather than the minor operation. The proof exploits the fact that triangulations have bounded face size and that edge‑contractions preserve planarity while gradually reducing the number of faces; a careful analysis shows that once the tree‑width exceeds a threshold, the graph must contain a sufficiently large “triangulated core” that can be contracted onto H.

Second, the paper provides a forbidden‑contraction characterization of 𝒞. The authors identify a finite set 𝔽 of minimal graphs that are not in 𝒞, and they prove that a planar graph belongs to 𝒞 if and only if it does not contain any member of 𝔽 as a contraction. This result gives a clean, combinatorial description of the boundary of 𝒞 and implies that membership in 𝒞 is decidable by checking a constant‑size obstruction list.

Armed with these structural insights, the authors design an algorithm that works in two phases. In the first phase the tree‑width of the input graph G is estimated (or computed exactly when feasible). If the tree‑width is at least c_H, the algorithm immediately answers “YES” by invoking the large‑tree‑width theorem. If the tree‑width is smaller, a tree‑decomposition of width < c_H is constructed, and a dynamic‑programming routine is run over this decomposition. The DP state records, for each bag, a partial mapping of the bag’s vertices to vertices of H together with information about which edges have already been contracted. Transition rules enforce the consistency of these mappings with the contraction operations. Because the width is bounded by the constant c_H, the DP runs in time polynomial in |V(G)|, with a multiplicative factor that depends only on |V(H)| (through the number of possible partial mappings).

The overall running time is therefore f(|V(H)|)·|V(G)|^{O(1)}. The function f captures the combinatorial explosion of possible mappings within a bag of size c_H and is independent of the size of G. In practice, for targets H that are modestly sized triangulations or their contractions, f is modest (often exponential in |V(H)| but with a small base), making the algorithm practical for large planar inputs.

The paper concludes with several directions for future work. One line of inquiry is to extend the approach beyond the class 𝒞, seeking other families of planar graphs that admit a similar separation of the complexity into an H‑dependent factor and a polynomial in |G|. Another direction is to investigate non‑planar graphs, where tree‑width alone may not guarantee the existence of a contraction, and to explore whether analogous “large‑width ⇒ forced contraction” theorems can be proved under additional topological constraints. Finally, the authors suggest leveraging the forbidden‑contraction set 𝔽 for related problems such as graph editing, pattern matching, and parameterized algorithms for contraction‑based graph modification.

In summary, the work provides a deep structural understanding of which planar graphs can be efficiently contracted to a given target, establishes a tight connection between tree‑width and contraction containment for the class of triangulation‑derived graphs, and delivers a practically efficient algorithm whose running time isolates the dependence on the target graph into a separate, well‑understood factor.