arXiv subjects

Seokbeom Kim

Publications and source records attributed to Seokbeom Kim.

5 recordsLinked to original sources

arXiv2026

An FPT algorithm for cycle rank on semi-complete digraphs

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph~$G$ and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^{w})} n^2$ determine whether $G$ has cycle rank at most~$w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^k)}n$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the \textsc{Minimum Feedback Arc Set} problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, when a cycle rank decomposition of depth $w\ge 1$ is given.

cs.DS↗

arXiv2026

On the chromatic number of the union of comparability graphs

Resolving in a strong sense a problem of Gyárfás on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs. We also show that the chromatic number can be replaced by the fractional chromatic number or $\frac{|V(G)|}{α(G)}$.

math.CO↗

arXiv2026

On a weaker notion of cross $t$-intersecting families

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1, \ldots, F_\ell \in \mathcal{F}$ and $F_1', \ldots, F_\ell' \in \mathcal{F}'$, then $\lvert \mathcal{F} \rvert \cdot \lvert \mathcal{F}' \rvert \leq \binom{n-t}{k-t} \binom{n-t}{k'-t}$, provided that $n$ is sufficiently large. This extends a celebrated theorem of Pyber for large $n$, which determines the tight upper bound for the product of the sizes of cross $1$-intersecting families.

math.CO↗

arXiv2025

The structure of $Δ(1, 2, 2)$-free tournaments

We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $Δ(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic triangle by substituting a two-vertex tournament for two of its vertices. In this paper, we show that tournaments excluding $Δ(1, 2, 2)$ as a subtournament are either isomorphic to one of three small tournaments, obtained from a transitive tournament by reversing edges in vertex-disjoint directed paths, or obtained from a smaller tournament with the same property by applying one of two operations. In particular, one of these operations creates a homogeneous set that induces a subtournament isomorphic to one of three fixed tournaments, and the other creates a homogeneous pair such that their union induces a subtournament isomorphic to a fixed tournament. As an application of this result, we present an upper bound for the chromatic number, a lower bound for the size of a largest transitive subtournament, and a lower bound for the number of vertex-disjoint cyclic triangles for such tournaments. The bounds that we present are all best possible.

math.CO↗

arXiv2025

An efficient algorithm for $\mathcal{F}$-subgraph-free Edge Deletion on graphs having a product structure

Given a family $\mathcal{F}$ of graphs, a graph is \emph{$\mathcal{F}$-subgraph-free} if it has no subgraph isomorphic to a member of $\mathcal{F}$. We present a fixed-parameter linear-time algorithm that decides whether a planar graph can be made $\mathcal{F}$-subgraph-free by deleting at most $k$ vertices or $k$ edges, where the parameters are $k$, $\lvert \mathcal{F} \rvert$, and the maximum number of vertices in a member of $\mathcal{F}$. The running time of our algorithm is double-exponential in the parameters, which is faster than the algorithm obtained by applying the first-order model checking result for graphs of bounded twin-width. To obtain this result, we develop a unified framework for designing algorithms for this problem on graphs with a ``product structure.'' Using this framework, we also design algorithms for other graph classes that generalize planar graphs. Specifically, the problem admits a fixed-parameter linear time algorithm on disk graphs of bounded local radius, and a fixed-parameter almost-linear time algorithm on graphs of bounded genus. Finally, we show that our result gives a tight fixed-parameter algorithm in the following sense: Even when $\mathcal{F}$ consists of a single graph $F$ and the input is restricted to planar graphs, it is unlikely to drop any parameters $k$ and $\lvert V(F) \rvert$ while preserving fixed-parameter tractability, unless the Exponential-Time Hypothesis fails.

cs.DM↗