A Polynomial Kernel for Vertex Deletion to the Scattered Class of Proper Interval Graph and Trees

Abstract: Vertex deletion to hereditary graph class is well-studied in parameterized complexity. Vertex deletion to the scattered graph classes has gained attention in recent years. In this paper, we consider (Proper-Interval, Tree)-Vertex Deletion, the input to which is an undirected graph $G = (V, E)$ and an integer $k$. The goal is to pick a set $X \subseteq V(G)$ of at most $k$ vertices such that $G - X$ is a simple graph and every connected component of $G - X$ is a proper interval graph or a tree. When parameterized by the solution size $k$, (Proper-Interval, Tree)-Vertex Deletion has been proved to be fixed-parameter tractable by Jacob et al. [JCSS-2023, FCT-2021]. In this paper, we consider this problem from the perspective of polynomial kernelization. We provide a first nontrivial polynomial kernel for (Proper-Interval, Tree)-Vertex Deletion, with $O(k<sup>{33})$ vertices.

• The paper establishes the first nontrivial polynomial kernel for a scattered-class vertex-deletion problem where both target classes have infinite forbidden induced-subgraph families, producing an instance with O(k^33) vertices.
• The kernel combines a 7-approximation, sunflower-based obstruction hitting, expansion lemmas, structural reductions, clique bypassing, and marking schemes to bound tree components by O(k^12) and proper interval components by O(k^33).
• The result broadens kernelization beyond finite-obstruction settings, while leaving open whether the exponent can approach O(k^6) and whether related deletion problems admit comparable kernels.

Problem and context

The paper studies the (Proper Interval, Tree)-Vertex Deletion (PITVD) problem: given an undirected graph $G$ and integer $k$, decide whether there is a set $X \subseteq V(G)$ with $|X| \le k$ such that every connected component of $G - X$ is a proper interval graph or a tree. Graphs whose components each lie in one of two hereditary classes are called scattered graphs, following the framework introduced by Ganian et al. and developed by Jacob et al., who established single-exponential FPT algorithms for several pairs including (Clique, Tree), (Interval, Tree), and (Proper Interval, Tree) (2605.02399).

The main contribution is a kernelization result:

PITVD admits a kernel with $\mathcal{O}(k^{33})$ vertices.

This is claimed to be the first nontrivial polynomial kernel for a $(\Pi_1, \Pi_2)$-Vertex Deletion problem in which both classes are characterized by infinite forbidden induced subgraph families. Prior work on polynomial kernels for scattered-class deletion was limited to (Clique, Tree)-Vertex Deletion — with kernels of $\mathcal{O}(k^5)$ vertices by Jacob et al., improved to $\mathcal{O}(k^4)$ by Tsur and to $\mathcal{O}(k^2)$ by Kumabe — plus folklore results that cover only finite forbidden families. In the (Clique, Tree) case one target class (cluster graphs) has a finite obstruction set; here neither proper interval graphs nor trees do, so the marking arguments must handle obstructions of unbounded size, namely holes of length at least 7 and (claw, triangle)-pairs within a single component.

The paper uses the known characterization: a simple graph is a (prop-int, tree)-graph if and only if it contains no net, no tent, no hole as an induced subgraph, and has no connected component containing both a claw and a triangle. Because some reduction rules introduce parallel edges, the algorithm works with multigraphs where edge multiplicity is capped at 2.

Overall structure of the kernelization

The algorithm proceeds in three phases. First, it invokes the 7-approximation algorithm of Jacob et al. to obtain a set $k$0 with $k$1 a (prop-int, tree)-graph; if $k$2 the instance is rejected. A sunflower-based lemma then produces a set $k$3 of size at most $k$4 such that hitting all nets, tents, and holes $k$5, $k$6, $k$7 inside $k$8 suffices to hit all such small obstructions globally. This reduces the remaining task to large obstructions: holes of length at least 7 and (claw, triangle)-pairs. The rest of the kernel bounds, separately, the vertices lying in tree components of $k$9 ($X \subseteq V(G)$0) and in proper interval components ($X \subseteq V(G)$1).

A suite of general preprocessing rules establishes structural guarantees: removal of components already in the target class; capping edge multiplicity at 2; a Buss-style rule deleting a vertex incident to $X \subseteq V(G)$2 double-edges; contraction of degree-2-tails to length 1; shortening degree-2-overbridges to length 4 via an edge-addition operation whose safeness rests on a lemma showing claws, nets, tents, and double-edges are preserved under the operation; pruning pendant trees to at most 5 vertices while keeping a nearest claw; and keeping at most 3 pendant trees per cut vertex. Two closure lemmas underpin these rules' correctness: attaching a degree-2-tail at a pendant vertex preserves the class, and subdividing an edge of an induced path in a proper interval graph preserves proper intervalness.

Bounding the tree components

For $X \subseteq V(G)$3, the analysis partitions vertices into $X \subseteq V(G)$4 and its complement, further splitting complement vertices into $X \subseteq V(G)$5-connecting vertices $X \subseteq V(G)$6, $X \subseteq V(G)$7-critical vertices, and $X \subseteq V(G)$8-hooks carrying "hanger" pendant trees. A reduction rule deletes hangers attached to bad $X \subseteq V(G)$9-hooks, justified by a claim that a good hook separates the bad hook from every relevant vertex of $|X| \le k$0, so any solution can be locally modified without increasing size.

Degree bounding at vertices of $|X| \le k$1 uses a flower argument: if $|X| \le k$2 contains a $|X| \le k$3-flower of order $|X| \le k$4, then $|X| \le k$5 belongs to every solution of size at most $|X| \le k$6 (via König's theorem applied to triplets of triangles yielding $|X| \le k$7 pairwise disjoint closest claw-triangle pairs). Otherwise, a set $|X| \le k$8 of at most $|X| \le k$9 vertices intersects every cycle, net, and pairwise-intersecting (claw, triangle)-pair through $G - X$0. An auxiliary bipartite graph between $G - X$1 and the components adjacent to $G - X$2 is then processed with the new $G - X$3-Expansion Lemma (Fomin et al.'s generalization requiring neither $G - X$4 nor absence of isolated vertices) with $G - X$5. The resulting reduction rule replaces edges from $G - X$6 to saturated components by double-edges to $G - X$7; its safeness proof shows that any optimal solution either contains $G - X$8 or can be strictly shrunk while containing $G - X$9. Consequently $\mathcal{O}(k^{33})$0, and counting leaves, critical vertices, good hooks, bounded pendant trees, and shortened degree-2-paths yields $\mathcal{O}(k^{33})$1.

Bounding the proper interval components

For $\mathcal{O}(k^{33})$2, the paper fixes a proper interval ordering and the associated clique partition $\mathcal{O}(k^{33})$3, exploiting the property that each clique's neighborhood lies in the two adjacent cliques. Three stages bound the structure:

Components. An auxiliary bipartite graph between $\mathcal{O}(k^{33})$4 and the components of $\mathcal{O}(k^{33})$5 is reduced with the classical Expansion Lemma at $\mathcal{O}(k^{33})$6: if there are at least $\mathcal{O}(k^{33})$7 components, a 3-expansion forces $\mathcal{O}(k^{33})$8 into every solution, since a vertex outside the solution together with neighbors in three triangle-containing components creates a claw-triangle pair.

Cliques. If some $\mathcal{O}(k^{33})$9 meets $(\Pi_1, \Pi_2)$0 distinct cliques of one component, then $(\Pi_1, \Pi_2)$1 is forced into the solution (five consecutive untouched cliques yield a hole or a pairwise-intersecting claw-triangle pair). For cliques disjoint from $(\Pi_1, \Pi_2)$2, a combinatorial lemma shows any seven consecutive such cliques contain two vertex-disjoint triangles; hence a window of $(\Pi_1, \Pi_2)$3 consecutive cliques straddling $(\Pi_1, \Pi_2)$4 allows a bypass operation (deleting a middle clique $(\Pi_1, \Pi_2)$5 and adding edges between its neighbors), safe because solutions intersecting the window can be exchanged with minimum $(\Pi_1, \Pi_2)$6-separators. Together these give $(\Pi_1, \Pi_2)$7 cliques.

Vertices per clique. A marking scheme inspired by Cao et al.'s unit-interval vertex deletion kernel marks, per clique $(\Pi_1, \Pi_2)$8: for each subset $(\Pi_1, \Pi_2)$9 of size at most 3 and each function $\mathcal{O}(k^5)$0, the first and last $\mathcal{O}(k^5)$1 vertices matching the adjacency pattern; plus three families of markers relative to neighboring cliques $\mathcal{O}(k^5)$2 involving common/non-common neighborhoods of up to $\mathcal{O}(k^5)$3 boundary vertices of $\mathcal{O}(k^5)$4. Each clique retains at most $\mathcal{O}(k^5)$5 marked vertices, and unmarked vertices are deleted. Safeness is established by two replacement lemmas: if an unmarked vertex participates in a hole of length at least 7 in $\mathcal{O}(k^5)$6, then $\mathcal{O}(k^5)$7 still contains a hole or a claw-triangle pair; analogously for components containing both a claw and a triangle. These proofs are extensive case analyses over how the hole or claw intersects $\mathcal{O}(k^5)$8, $\mathcal{O}(k^5)$9, and the remainder $\mathcal{O}(k^4)$0.

Combining the bounds — $\mathcal{O}(k^4)$1 cliques times $\mathcal{O}(k^4)$2 vertices per clique gives $\mathcal{O}(k^4)$3, dominating $\mathcal{O}(k^4)$4 and $\mathcal{O}(k^4)$5 — yields the $\mathcal{O}(k^4)$6-vertex kernel.

Limitations and open questions

The exponent 33 is large and admittedly not tight; the authors state that improving it toward $\mathcal{O}(k^4)$7 should be possible but requires new techniques specifically for handling nets and tents. The kernel also relies on the existing 7-approximation and the single-exponential $\mathcal{O}(k^4)$8 FPT algorithm of Jacob et al.; whether the latter can be improved to $\mathcal{O}(k^4)$9 or $\mathcal{O}(k^2)$0 remains open. The authors further leave open whether their approach extends to (Interval, Forest) or (Chordal, Bipartite Permutation) deletion, and to deletion into three or more scattered classes.

Conclusion

The paper provides the first nontrivial polynomial kernel for vertex deletion to a scattered pair of graph classes where both classes have infinite forbidden induced subgraph families, achieving $\mathcal{O}(k^2)$1 vertices via a combination of approximation-guided decomposition, expansion-lemma-based degree reduction, separator-based clique bypassing, and a fine-grained marking scheme over proper interval orderings. The result significantly broadens the scope of kernelization for scattered-class deletion beyond finite-obstruction settings, though closing the gap between $\mathcal{O}(k^2)$2 and plausible lower bounds remains open.