Cliques or Trees Vertex Deletion

• The paper introduces the cliques-or-trees vertex deletion problem by unifying Cluster Vertex Deletion and Feedback Vertex Set approaches.
• It presents a novel quadratic kernel with O(k²) vertices and explicit reduction rules that bridge dense clique components and sparse tree components.
• The study demonstrates fixed-parameter tractability through efficient algorithms and detailed structural characterizations that guide practical applications.

Cliques-or-Trees Vertex Deletion is the problem of deciding, for a graph $G$ and an integer $k$, whether there exists a set $X \subseteq V(G)$ of size at most $k$ such that every connected component of $G-X$ is either a clique or a tree. In the language of deletion to scattered graph classes, it is the problem of deleting to $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$ for the pair consisting of cliques and trees, and it is explicitly presented as a hybrid of two fundamental parameterized problems: Cluster Vertex Deletion and Feedback Vertex Set (Jacob et al., 2024, Kumabe, 20 Sep 2025).

1. Formalization and parameterization

For two graph classes $\mathcal{G}_1,\mathcal{G}_2$, the scattered class is defined by

$$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$

The specific problem asks, given $G=(V,E)$ and $k\in\mathbb{N}$, whether there exists $k$0 with $k$1 such that for every component $k$2 of $k$3, $k$4 is a complete graph $k$5 with $k$6 or $k$7 is a tree (Jacob et al., 2024).

The quadratic-kernel study defines the associated structural parameter $k$8 as the minimum $k$9 such that there exists $X \subseteq V(G)$0, $X \subseteq V(G)$1, for which every connected component of $X \subseteq V(G)$2 is either a clique or a tree. In the same framework, $X \subseteq V(G)$3 denotes Cluster Vertex Deletion and $X \subseteq V(G)$4 denotes Feedback Vertex Set. The immediate inequalities

$X \subseteq V(G)$5

formalize the sense in which the parameter generalizes both dense-side and sparse-side deletion measures (Kumabe, 20 Sep 2025).

Two basic properties are recorded for $X \subseteq V(G)$6. It is monotone under vertex deletion: for any $X \subseteq V(G)$7, $X \subseteq V(G)$8. It is also additive under disjoint union: for disjoint graphs $X \subseteq V(G)$9,

$k$0

The same work notes that it is not hard to see that graphs with bounded cliques-or-trees vertex deletion number have bounded clique-width, and uses this to position the parameter as a more specific alternative to clique-width (Kumabe, 20 Sep 2025).

2. Obstruction viewpoint and structural characterization

A central structural fact is the forbidden-induced-subgraph characterization of the target class. Let $k$1 be the class of simple graphs whose components are cliques or trees. Then

$k$2

Here a paw has vertices $k$3 forming a triangle and a pendant vertex $k$4 adjacent to $k$5, while a diamond is $k$6 minus one edge (Jacob et al., 2024).

This obstruction set explains why the problem is not simply the disjoint union of cluster deletion and forest deletion. On the clique side, induced $k$7 configurations are the core obstruction for cluster graphs; on the tree side, cycles are the core obstruction for forests. In the scattered setting, however, the relevant obstructions are those that certify that a connected component is neither a clique nor a tree. The 2024 kernel paper emphasizes that obstructions for the two classes may reside in different parts, and that a solution may “separate” obstructions rather than hit them. It therefore states explicitly that naively combining kernels for Cluster Vertex Deletion and Feedback Vertex Set does not work (Jacob et al., 2024).

The parameter inequalities $k$8 and $k$9 provide the formal relation to these neighboring deletion problems. A cluster-deletion solution is automatically valid because a disjoint union of cliques satisfies the target condition, and a feedback vertex set is automatically valid because a forest has all components as trees (Kumabe, 20 Sep 2025).

3. Fixed-parameter tractability and the first polynomial kernel

Before quadratic kernelization, the problem already had fixed-parameter algorithms. The 2024 kernel paper states that an $G-X$0 algorithm is known for Cliques-or-Trees Vertex Deletion, and then presents the first non-trivial polynomial kernel for the problem, with $G-X$1 vertices (Jacob et al., 2024).

The kernel starts from a 4-approximation due to Jacob–Majumdar–Raman, producing a deletion set $G-X$2 with $G-X$3, or returning NO otherwise. It maintains the invariant that every remaining component in $G-X$4 has at least one neighbor in $G-X$5. The reduction system combines isolated component removal, an edge-multiplicity cap, pendant-vertex bounding, reductions for degree-2 tails and induced degree-2 overbridges, a 2-expansion rule for bounding the number of clique components, a clique-marking rule denoted Mark-Clique-K, tree-side leaf trimming, pendant tree contraction, a $G-X$6-flower rule, and a new 4-Expansion Lemma for tree-side adjacency (Jacob et al., 2024).

The clique side is controlled by bounding the number of clique components of $G-X$7 of size at least $G-X$8, and then bounding the size of each such clique via the marking rule. The tree side is controlled by first bounding the number of vertices adjacent to $G-X$9, and then using the local path and pendant reductions to bound the size of the forest part. The resulting estimates are explicit: after the 2-expansion rule, the number of clique components of size at least $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$0 is at most $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$1; on the forest side, if $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$2 denotes the vertices in tree components of $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$3, then

$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$4

Putting these together with the marking bound for clique components yields

$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$5

for the reduced instance (Jacob et al., 2024).

Historically, this was described as the first non-trivial polynomial kernel for deletion to a scattered pair of graph classes. Its significance lies not only in the $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$6 bound, but also in the introduction of techniques that explicitly mediate between dense components and sparse components through a common boundary set $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$7 (Jacob et al., 2024).

4. Quadratic kernelization

The 2025 work improves the kernel size to $\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$8 vertices and gives the explicit post-reduction bound

$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)$9

Its main theorem states that Cliques or Trees Vertex Deletion admits a kernel with $\mathcal{G}_1,\mathcal{G}_2$0 vertices (Kumabe, 20 Sep 2025).

The kernelization is organized around a degree-and-density partition of the vertex set. Using the threshold constant $\mathcal{G}_1,\mathcal{G}_2$1 and neighbor-density threshold $\mathcal{G}_1,\mathcal{G}_2$2, the paper defines

$\mathcal{G}_1,\mathcal{G}_2$3

$\mathcal{G}_1,\mathcal{G}_2$4

$\mathcal{G}_1,\mathcal{G}_2$5

Two structural lemmas then separate the tree side from the clique side: if $\mathcal{G}_1,\mathcal{G}_2$6 and $\mathcal{G}_1,\mathcal{G}_2$7, then $\mathcal{G}_1,\mathcal{G}_2$8 lies in a tree component of $\mathcal{G}_1,\mathcal{G}_2$9; if $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$0 and $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$1, then $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$2 lies in a clique component of $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$3 (Kumabe, 20 Sep 2025).

The tree-side reduction suite is described as closely following Thomassé’s quadratic kernel for Feedback Vertex Set. It uses Gallai’s theorem to obtain either a $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$4-flower of order $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$5 or a blocker $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$6 of size at most $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$7, together with rules for many non-tree components, tree pruning, and a 2-expansion reduction on tree components. One conclusion is formalized as: after exhaustive application, all vertices outside $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$8 have degree at most $$\mathrm{Scatter}(\mathcal{G}_1,\mathcal{G}_2)=\{\,G:\text{ every connected component of }G\text{ lies in }\mathcal{G}_1\cup\mathcal{G}_2\,\}.$$9; equivalently, $G=(V,E)$0 is empty (Kumabe, 20 Sep 2025).

The clique-side reduction suite begins from a 4-approximation $G=(V,E)$1 with $G=(V,E)$2. It applies a correspondence rule for large-dense vertices that appear inside tree components of $G=(V,E)$3, a 2-expansion reduction on clique components, and a packing analysis of induced $G=(V,E)$4 subgraphs inside $G=(V,E)$5. The maximal family $G=(V,E)$6 of induced $G=(V,E)$7s with pairwise intersection of size at most $G=(V,E)$8 is bounded by $G=(V,E)$9, hence $k\in\mathbb{N}$0. Multi-edges are handled by Buss rules for Vertex Cover, and the remainder of the large-dense region is organized into extended clique-modules. The final bound on the large-dense side is

$k\in\mathbb{N}$1

On the tree side, the paper obtains

$k\in\mathbb{N}$2

Combining these with the bounds on $k\in\mathbb{N}$3 and on clique components outside $k\in\mathbb{N}$4 yields the final estimate

$k\in\mathbb{N}$5

for the reduced instance (Kumabe, 20 Sep 2025).

A notable methodological point is that the paper presents this result as closing the gap with Feedback Vertex Set, which also admits an $k\in\mathbb{N}$6-vertex kernel. At the same time, it remarks that $k\in\mathbb{N}$7 generalizes both $k\in\mathbb{N}$8 and $k\in\mathbb{N}$9, while still being more specific than clique-width (Kumabe, 20 Sep 2025).

5. Generalizations and algorithmic consequences

A broader scattered-class result was obtained for $k$00-Vertex Deletion. That problem asks for a set $k$01 of size at most $k$02 such that $k$03 is a simple graph and every connected component of $k$04 is a proper interval graph or a tree. The main kernelization theorem gives a polynomial kernel with $k$05 vertices (Jacob et al., 4 May 2026).

For Cliques-or-Trees Vertex Deletion, this yields an immediate corollary because cliques are proper interval graphs. The paper states: since the class of cliques is a subclass of proper interval graphs, the kernelization for $k$06-Vertex Deletion yields a polynomial kernel of size $k$07 for $k$08-Vertex Deletion. It also notes that the literature has stronger kernels specifically tailored to $k$09-Vertex Deletion, down to $k$10 (Jacob et al., 4 May 2026).

The parameter $k$11 also has consequences beyond kernelization. The 2025 quadratic-kernel paper proves that Longest Cycle becomes fixed-parameter tractable when parameterized by the cliques or trees vertex deletion number. Its algorithm runs in time

$k$12

The high-level method guesses the order in which a candidate cycle visits the vertices of a modulator $k$13, labels top $k$14 components per ordered pair $k$15 by the value $k$16, and reduces the remainder to per-component instances of Longest Disjoint Terminal Paths. Clique components are handled in $k$17 time via matching-based feasibility, while tree components are handled in $k$18 time via dynamic programming over a rooted tree (Kumabe, 20 Sep 2025).

This combination of kernelization and downstream parameterized algorithms is one reason the 2025 work explicitly “initiates the study of the cliques or trees vertex deletion number as a structural parameter” (Kumabe, 20 Sep 2025).

6. Relation to adjacent deletion problems

The clique side of the subject is tightly connected to Cluster Vertex Deletion. A cluster graph is a disjoint union of cliques, equivalently a $k$19-free graph. For $k$20-free graphs, the complexity of Cluster Vertex Deletion and Connected Cluster Vertex Deletion is completely characterized: both are polynomially solvable on $k$21-free graphs if $k$22 is an induced subgraph of $k$23, and NP-complete otherwise; in the NP-complete cases there is no $k$24-time algorithm on $k$25-free $k$26-vertex graphs unless ETH fails (Le et al., 2024). On the parameterized side, Cluster Vertex Deletion has an $k$27-time branching algorithm based on the auxiliary graph $k$28, which encodes induced $k$29s through a pivot vertex $k$30 as edges in a local vertex-cover instance (Boral et al., 2013). On chordal graphs, the weighted problem is polynomial-time solvable by dynamic programming over clique trees and a reduction to the maximization of a supermodular function (Cao et al., 22 Apr 2026).

The tree side has its own deletion landscape. The problem of deleting vertices or edges to obtain a full binary tree is NP-hard in both versions, and both problems are fixed-parameter tractable by the standard parameter $k$31. The same work contrasts this with the classical fact that deleting edges to obtain a forest or a tree is equivalent to computing a minimum spanning tree, which is polynomial-time solvable (Dayal et al., 2019).

These neighboring results clarify the position of Cliques-or-Trees Vertex Deletion. It inherits the clique-oriented behavior of Cluster Vertex Deletion and the cycle-oriented behavior of tree-deletion problems, but its scattered-class target creates genuinely mixed instances in which some components are optimally resolved on the dense side and others on the sparse side. The kernelization papers make this point explicit by using obstruction packings, expansion lemmas, clique-side marking, Gallai-style cycle control, and component correspondences that are specific to the hybrid target rather than to either constituent problem in isolation (Jacob et al., 2024, Kumabe, 20 Sep 2025).