In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in H-free graphs. For Tight b-Chromatic Number in H-free graphs, we develop a general technique to determine new graphs H, for which the problem is polynomial-time solvable, and we also determine new graphs H, for which the problem is still NP-complete. We show, for the first time, the existence of a graph H such that in H-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets B and R such that the edges between B and R form a matching, that is, every vertex in B has at most one neighbour in R, and vice versa. If for some integer d ≥ 1, we allow every vertex in B to have at most d neighbours in R, and vice versa, we obtain the more general problem d-Cut. It is known that d-Cut is NP-complete for every d ≥ 1. However, for claw-free graphs, it is only known that d-Cut is polynomial-time solvable for d = 1 and NP-complete for d ≥ 3. We resolve the missing case d = 2 by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every d ≥ 2, d-Cut, restricted to claw-free graphs of maximum degree p, is constant-time solvable if p ≤ 2d+1 and NP-complete if p ≥ 2d+3. Moreover, in the former case, we can find a d-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to S_{1^t,𝓁}-free graphs where S_{1^t,𝓁} is the graph obtained from a star on t+2 vertices by subdividing one of its edges exactly 𝓁 times.

An induced packing of cycles in a graph is a set of vertex-disjoint cycles with no edges between them. We generalise the classic Erdős-Pósa theorem to induced packings of cycles. More specifically, we show that there exists a function f (k) = O (k log k ) such that for every positive integer k, every graph G contains either an induced packing of k cycles or a set X of at most f (k ) vertices such that the closed neighbourhood of X intersects all cycles in G. Our proof is constructive and yields a polynomial-time algorithm finding either the induced packing of cycles or the set X. Furthermore, we show that for every positive integer d, if a graph G does not contain two cycles at distance more than d, then G contains sets X1, X2 ⊆ V (G ) with |X1| ≤ 12(d + 1) and |X2| ≤ 12 such that, after removing the ball of radius 2d around X1 or the ball of radius 3d around X2, the resulting graphs are forests.

International audience

Abstract. For each [Formula: see text], we construct a finite set [Formula: see text] of multigraphs such that for each graph [Formula: see text] of girth at least 5 obtained from a multigraph [Formula: see text] by subdividing each edge at least two times, [Formula: see text] has twin-width at most [Formula: see text] if and only if [Formula: see text] has no minor in [Formula: see text]. This answers a question of Bergé, Bonnet, and Déprés asking for the structure of graphs [Formula: see text] such that each long subdivision of [Formula: see text] has twin-width 4. As a corollary, we show that the [Formula: see text] grid has twin-width 4, which answers a question of Schidler and Szeider.

Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$. The \textsc{$(p,r,\mathcal{F})$-Packing} problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $i,j\in \{1, \ldots, k\}$, the distance between $V(H_i)$ and $V(H_j)$ in $G$ is larger than $r$. We show that for every fixed nonnegative integers $p,r$ and every fixed nonempty finite family $\mathcal{F}$ of connected graphs, the \textsc{$(p,r,\mathcal{F})$-Covering} problem with $p\leq2r+1$ and the \textsc{$(p,r,\mathcal{F})$-Packing} problem with $p\leq2\lfloor r/2\rfloor+1$ admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size $k$. We obtain the same kernels for their annotated variants. As corollaries, we prove that \textsc{Distance-$r$ Vertex Cover}, \textsc{Distance-$r$ Matching}, \textsc{$\mathcal{F}$-Free Vertex Deletion}, and \textsc{Induced-$\mathcal{F}$-Packing} for any fixed finite family $\mathcal{F}$ of connected graphs admit almost linear kernels on every nowhere dense class of graphs and linear kernels on every class of graphs with bounded expansion. Our results extend the results for \textsc{Distance-$r$ Dominating Set} by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for \textsc{Distance-$r$ Independent Set} by Pilipczuk and Siebertz (EJC 2021).

For a non-negative integer $$\ell $$ , the $$\ell $$ -leaf power of a tree T is a simple graph G on the leaves of T such that two vertices are adjacent in G if and only if their distance in T is at most $$\ell $$ . We provide a polynomial kernel for the problem of deciding whether we can delete at most k vertices to make an input graph a 3-leaf power of some tree. More specifically, we present a polynomial-time algorithm for an input instance (G, k) for the problem to output an equivalent instance $$(G',k')$$ such that $$k'\leqslant k$$ and $$G'$$ has at most $$O(k^{14})$$ vertices.

For a family of graphs $$\mathcal {F}$$ , Weighted $$\mathcal {F}$$ -Deletion is the problem for which the input is a vertex weighted graph $$G = (V, E)$$ and the goal is to delete $$S \subseteq V$$ with minimum weight such that $$G \setminus S \in \mathcal {F}$$ . Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when $$\mathcal {F}$$ is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest.

Bonnet et al. [ J. ACM, 69 (2022), 3] introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdös--Rényi random graph $G(n,p)$ with $1/n\leq p\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Last, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.

For a non-negative integer 𝓁, a graph G is an 𝓁-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most 𝓁. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ⊆ V(G) of size at most k such that G\S is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G',k') such that k'≤ k and G' has at most O(k^14) vertices.