A polynomial kernel for vertex deletion into bipartite permutation graphs

TL;DR

The paper studies the Bipartite Permutation Vertex Deletion problem, where one seeks to delete at most $k$ vertices to obtain a bipartite permutation graph (a NP-hard, FPT problem). It develops a polynomial kernel of size $O(k^{62})$ by combining a forbidden-subgraph characterization with a sunflower-based compression of small structures, followed by reduction rules that bound both isolated and non-isolated vertices after removing a modulator $T$. The analysis introduces explicit polynomial bounds $\\delta(k)$, $\\epsilon(k)$, and $\phi(k)$ to control different vertex groups and shows how they culminate in a kernel of size $\xi(k)=O(k^{62})$. This work advances kernelization techniques for vertex-deletion problems into graph classes and demonstrates a concrete, scalable pre-processing bound for this graph class.

Abstract

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $\ell_1$ and $\ell_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In the the bipartite permutation vertex deletion problem we ask for a given $n$-vertex graph, whether we can remove at most $k$ vertices to obtain a bipartite permutation graph. This problem is NP-complete but it does admit an FPT algorithm parameterized by $k$. In this paper we study the kernelization of this problem and show that it admits a polynomial kernel with $O(k^{62})$ vertices.

Figures (11)

• Figure 1: Forbidden structures for bipartite permutation graphs.
• Figure 2: Embedding of a bipartite permutation graph $(U,W,E)$ into a strip satysfying the adjacency and the enclosure properties.
• Figure 3: The set $T$ and the graph $G-T$. The set $C$ contains non-isolated vertices in $G-T$.
• Figure 4: An induced subpathpath of $P$ ordered by strong ordering. Note that there is no way to place a vertex $u$ such that $\{w_0, w_2, w_4, w_6\} \subseteq N(u)$.
• Figure 5: Forbidden induced cycle $S$ and the vertex $v'$. The vertices $u,w\in T$ are depicted in red. Vertex $v$ is irrelevant because a vertex $v'$ would also form a forbidden induced cycle.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2: Spinrad, Brandstädt, Stewart SBS87
• Lemma 3: Sunflower lemma DBLP:series/txtcs/FlumG06
• Lemma 4: DBLP:journals/siamdm/FominSV13
• Lemma 5
• proof
• Lemma 6: The Path Lemma
• proof
• Claim 6.1
• proof