Amir Nikabadi

Paweł Rafał Bieliński, Jadwiga Czyżewska, Martin Milanič et al.

A $P_4$ is a chordless path on four vertices. A diamond is a graph obtained from a clique of size four by removing one edge of the clique. A paw is a graph obtained from a clique of size four by removing two adjacent edges of the clique. We prove that for a graph $H$, the class of graphs with no induced subdivision of $H$ has bounded clique-width if and only if $H$ is an induced subgraph of $P_4$, the paw, or the diamond. This answers a~question of Dabrowski, Johnson, and Paulusma.

Amir Nikabadi, Eva Rotenberg, Lasse Wulf

When a graph $G$ admits a vertex $v$ that is contained in all its longest paths, we call $v$ a Gallai vertex. These are named after Gallai, who in 1966 asked the question if it is true that every connected graph contains such a vertex. This was soon answered in the negative by Walther and Zamfirescu, who presented a graph in which every vertex is omitted by some longest path of the graph. In spite of its long history, the Gallai Vertex Problem, i.e. determining whether a graph has a Gallai vertex, was until now neither known to be NP- nor co-NP-hard. In this work, we show something much stronger, as we completely settle the computational complexity of determining whether a graph has a Gallai vertex: we show that it is complete for the complexity class $Θ_2^p = \text{P}^{\text{NP}[\log n]}$. This class, also known as parallel access to NP, is a complexity class larger than NP situated just below the class $Σ^p_2$ in Stockmeyer's polynomial hierarchy. In more generality, the longest path transversal number of a connected graph is the minimum size of a set of vertices that intersects all its longest paths. I.e. if the graph has a Gallai vertex, its longest path transversal number is $1$. Thus, as a consequence of our theorem, the longest path transversal number of a graph cannot be approximated in polynomial time by a factor better than 2, unless $\text{P} = \text{NP}$. In fact, using related techniques, we show a strengthening of this result: For any constant $C$, if there is a graph with longest path transversal number $C$, then there is no polynomial time algorithm for approximating the longest path transversal number by a factor better than $C$, unless $\text{P} = \text{NP}$. In particular, this excludes approximation by a factor below $3$. Similar results hold for the longest cycle transversal.

Amir Nikabadi

For a graph $G$, the \emph{equitable chromatic number} of $G$, denoted by $χ_e(G)$, is the smallest integer $k$ such that $G$ admits a proper $k$-coloring whose color classes differ in size by at most one. We prove that for every $ζ>41/2$, there exists a constant $c=c(ζ)\in\mathbb{N}$ such that every bipartite graph $G$ with maximum degree $Δ(G)\ge c$ and $|V(G)|\ge ζΔ(G)$ satisfies $χ_e(G)\le \left\lceilΔ(G)/2\right\rceil+1$. The leading term $Δ(G)/2$ in this bound is best possible for upper bounds stated solely in terms of $Δ(G)$ for bipartite graphs. Our proof yields an $O(|V(G)|^2)$-time algorithm for constructing such a coloring.