Kernelization Dichotomies for Hitting Subgraphs under Structural Parameterizations

TL;DR

The paper investigates kernelization for the H-Subgraph Hitting and H-Induced Subgraph Hitting problems under structural parameters, introducing ved^+ and bed^+ as generalized elimination-distance measures. It establishes a sharp dichotomy: for fixed H that is biconnected, the problems admit polynomial kernels parameterized by a modulator to a simple class if and only if H is a clique, revealing a fundamental barrier beyond cliques. The authors also provide a polynomial kernel for K_t-Subgraph Hitting under bed^+-type constraints and develop a suite of tools (marking, chunks, conflicts, blocking sets) to bound kernel size via mmbs_t, plus algorithms to compute bed^+ roots and optimal solutions in bounded-bed instances. Hardness results show that, when H is not a clique, substantial kernelization is unlikely under standard complexity assumptions, and the induced-subgraph variant exhibits different behavior from minors. The work advances a cohesive framework for kernelization under distance-to-triviality, clarifying when efficient compression is possible and highlighting the distinct landscape for hitting subgraphs versus minors.

Abstract

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

Figures (5)

• Figure 1: In this example we consider that $H=K_4$, and we denote by $C_1$ and $C_2$ the two connected components of $G$ (where $v_1 \in C_1$). Observe that $\mathcal{T}\xspace=(T_1,T_2)$ is a root of the depicted graph with $T_1 = \{v_1,v_2,v_3,v_4\}$ and $T_2 = \{u_1,u_2\}$. We have $C(v_1)=\{v_1\}$ and $C(v_2)=\{v_2,w_1,w_2,w_3\}$. Finally, taking $T'_1 = T_1 \cup \{w_1\}$ and $\mathcal{T}\xspace' = \{T_1',T_2\}$ would not be a root as $T'_1$ is not a root of $G[C_1]$.
• Figure 2: Main steps of the kernel. In this example $\mathcal{T}\xspace = \{T_1,T_2\}$ (edges inside $T_i$ are in bold, and dotted edges cannot exist), and there exists a non-marked vertex $v$, implying that the pending component $C(v)$ will be removed.
• Figure 3: Example of a call to mark$(\mathcal{T}\xspace,N,X',c,N',M')$ and of a $(\mathcal{T}\xspace,N,X',c,N',M')$-part $(V_i,N_i)$ (with in particular ${\sf conf}\xspace^t_{X'}(M' \cup C (V_i) \cup N' \cup N_i) > 0$). The nine vertices of $V(\mathcal{T}\xspace)$ are depicted as black filled circles.
• Figure 4: Setting of \ref{['lemma:markingalgo']}. The set $M(\mathcal{T}\xspace,N,G,X)$ is denoted by $M$. The ten vertices of $V(\mathcal{T}\xspace)$ are depicted as black filled circles and $\mathcal{P}\xspace^\star=\{(V_i,N_i),(V_j,N_j)\}$.
• Figure 7: Two chains of length three incomparable with respect to the (induced) subgraph relation. In both graphs, it can be verified that the set of four red thicker edges is a minimal blocking set.

Theorems & Definitions (54)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Definition 5: root and pending component
• Definition 7
• Theorem 8
• Theorem 9
• Definition 10: projection
• Definition 11