A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.

Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the Matching Cut problem and some variants. Few other results, mostly on Longest Path and some generalizations of Matching Cut, have also been developed. However, little success has been seen in enumeration versions of Vertex Cover and Feedback Vertex Set, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with 2k vertices for Enum Vertex Cover, where we wish to list all solutions with at most k vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with 𝒪(k²) vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with 𝒪(k³) vertices and edges for Enum Feedback Vertex Set; the proof is inspired by some ideas of Thomassé [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the q-expansion technique.

Musical arrangements have recently been considered from an algorithmic point of view. Demaine and Moses, in 2017, introduced three decision problems associated with musical arrangements. In these problems, the input is a score H consisting of n staves corresponding to the participating instruments and a parameter p, 0 ≤ p ≤ 1. The goal is to determine whether there is a subset H’ c H satisfying some properties, such that in each time unit p% of the input score H is played. In this paper, we take a step forward in this direction. We define more general versions of two of the original problems, which we call general consonant arrangement (con-arr) and general j -simultaneous notes ( j-notes ) with two additional parameters: s , a lower bound for the number of required staves in the solution, and k , a lower bound for the number of time units in which p% of the input score is played. We state the name of the problem followed by the parameter in parentheses to indicate which parameter is being maximized. We show that con-arr( k ) is MAXSNP-hard and that con-arr( s ) and j-notes(s) are not approximable within n 1_ε , for every ε > 0, unless P=NP. Let ∆ be the maximum number of staves that play simultaneously with any single stave at any moment in the music. If ∆ = 4, then con-arr(s) is MAXSNP-hard, because maximum degree 4 independent set is MAXSNP-complete, and it is polynomial-time 1/5-approximable. The j-notes (s) problem is O((∆ + 1) s .n 2 )-time solvable, i.e., it is in FPT with respect to the parameters ∆ and s. We also introduce two new problems we think are of musical interest: the arrangement for k instruments ( k -arrangement) and the time filling by instruments (fill-inst), which we prove to be hard. In particular, k -arrangement is hard even if each stave has exactly 2 notes. Also, k -arrangement is not polynomially approximable within a n 1 - ε factor, for ε > 0, unless P=NP. Finally, fill-inst is in FPT with respect to parameter τ, the maximum number of staves playing in each time, and in the size k of the solution. We prove that if P≠NP, then the best approximation ratio for fill-inst is Θ(log T) , where T is the number of non-silent times of an input.

In the graph coloring game , given a graph G and c colors, two players, Alice and Bob take turns by proper coloring the vertices of G, meaning adjacent vertices must have distinct colors. The game ends when there are no valid moves. Alice wins if all vertices are colored, and Bob wins otherwise. The game chromatic number x g (G) is the least integer k such that Alice has a winning strategy with k colors. We show that Xg ( G ) ≤ 2k - 3 for threshold graphs, Xg(G) ≤ 2k- 2 for the disjoint union of threshold graphs, and Xg(G) ≤ 2k- 1 for the disjoint union of split graphs, where k is the clique number of G. We also show that these bounds are tight and provide bounds for the variant where Bob starts.

A graph convexity on a finite graph G is a family C ⊆ 2 V(G) closed for intersections containing θ and V(G). The elements of C are said to be convex sets. Several graph convexities have been considered in the literature motivated by structural properties, analogies to geometry or applications. One of those is the cycle convexity, in which a set S ⊆ V(G) is convex if for every v ε G\S there is no cycle containing v in the subgraph induced by S U { v }. Several natural questions arise, when studying graph convexities. One of them is the problem of partitioning the vertex set of a graph on a given number k of convex sets. This problem has been previously addressed for different graph convexities. In this paper, we extend this line of research for the cycle convexity, by showing that the problem is NP-complete for every fixed k ≥ 2. On the positive side, we provide a linear-time algorithm for the case when the input graph is chordal.

Graph modification problems are computational tasks where the goal is to change an input graph G using operations from a fixed set, in order to make the resulting graph satisfy a target property, which usually entails membership to a desired graph class C. Some well-known examples of operations include vertex-deletion, edge-deletion, edge-addition and edge-contraction. In this paper we address an operation known as subgraph complement. Given a graph G and a subset S of its vertices, the subgraph complement G ⊕ S is the graph resulting from complementing the edge set of the subgraph induced by S in G. We say that a graph H is a subgraph complement of G if there is an S such that H is isomorphic to G⊕S. For a graph class C, the Subgraph complementation to C is the problem of deciding, for a given graph G, whether G has a subgraph complement in C. This problem has been studied and its complexity has been settled for many classes C such as H -free graphs, for various families H , and for classes of bounded degeneracy. In this work, we focus on classes of graphs of minimum/maximum degree upper/lower bounded by some value k. In particular, we answer an open question of Antony et al. [Information Processing Letters 188, 106530 (2025)], by showing that Subgraph complementation to C is NP-complete when C is the class of graphs of minimum degree at least k , if k is part of the input. We also show that Subgraph complementation to k -regular parameterized by k is fixed-parameter tractable.

A matching cut of a graph is a partition of its vertex set in two such that no vertex has more than one neighbor across the cut. The Matching Cut problem asks if a graph has a matching cut. This problem, and its generalization d-cut, has drawn considerable attention of the algorithms and complexity community in the last decade, becoming a canonical example for parameterized enumeration algorithms and kernelization. In this paper, we introduce and study a generalization of Matching Cut, which we have named Matching Multicut: can we partition the vertex set of a graph in at least $\ell$ parts such that no vertex has more than one neighbor outside its part? We investigate this question in several settings. We start by showing that, contrary to Matching Cut, it is NP-hard on cubic graphs but that, when $\ell$ is a parameter, it admits a quasi-linear kernel. We also show an $O(\ell^{\frac{n}{2}})$ time exact exponential algorithm for general graphs and a $2^{O(t \log t)}n^{O(1)}$ time algorithm for graphs of treewidth at most $t$. We then study parameterized enumeration aspects of matching multicuts. First, we generalize the quadratic kernel of Golovach et. al for Enum Matching Cut parameterized by vertex cover, then use it to design a quadratic kernel for Enum Matching (Multi)cut parameterized by vertex-deletion distance to co-cluster. Our final contributions are on the vertex-deletion distance to cluster parameterization, where we show an FPT-delay algorithm for Enum Matching Multicut but that no polynomial kernel exists unless NP $\subseteq$ coNP/poly; we highlight that we have no such lower bound for Enum Matching Cut and consider it our main open question.

For a fixed property (graph class) Π, given a graph G and an integer k, the Π-deletion problem consists in deciding if we can turn G into a graph with the property Π by deleting at most k edges. The Π-deletion problem is known to be NP-hard for most of the well-studied graph classes, such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others; even deletion to cacti is known to be NP-hard for general graphs. However, there is a notable exception: the deletion problem to trees is polynomial. Motivated by this fact, we study the deletion problem for some classes similar to trees, addressing in this way a knowledge gap in the literature. We prove that deletion to cacti is hard even when the input is a bipartite graph. On the positive side, we show that the problem becomes tractable when the input is chordal, and for the special case of quasi-threshold graphs we give a simpler and faster algorithm. In addition, we present sufficient structural conditions on the graph class Π that imply the NP-hardness of the Π-deletion problem, and show that deletion from general graphs to some well-known subclasses of forests is NP-hard.

We study the problem of reconfiguring one minimum $s$-$t$-separator $A$ into another minimum $s$-$t$-separator $B$ in some $n$-vertex graph $G$ containing two non-adjacent vertices $s$ and $t$. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming $A$ into $B$. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most $\ell$ jumps can transform $A$ into $B$ is an $\textsf{NP}$-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size $k$ of the minimum \stseps and when parameterized by the number of jumps $\ell$. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complete the picture by designing a kernel with $\mathcal{O}(\ell^2)$ vertices and edges for the length $\ell$ of the sequence as a parameter.

Given a graph G on n vertices and two integers k and d, the Contraction(vc) problem asks whether one can contract at most k edges to reduce the vertex cover number of G by at least d. Recently, Lima et al. [JCSS 2021] proved that Contraction(vc) admits an XP algorithm running in time f(d)⋅nO(d). They asked whether this problem is FPT under this parameterization. In this article, we prove that: (i) Contraction(vc) is W[1]-hard parameterized by k+d. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f(k+d)⋅no(k+d) for any function f. This answers negatively the open question stated in Lima et al. [JCSS 2021]. (ii) Contraction(vc) is NP-hard even when k=d. (iii) Contraction(vc) can be solved in time 2O(d)⋅nk−d+O(1). This improves the algorithm of Lima et al. [JCSS 2021], and shows that when k=d, Contraction(vc) is FPT parameterized by d (or by k).

International audience

A locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to be NP-hard even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set is W[1]-hard when parameterized by the size of a minimum clique cover. We present an exponential kernel for the distance to cluster parameterization and show that, unless NP ⊆ coNP/poly, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by either of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

An n-vertex graph is equitably k-colorable if there is a proper coloring of its vertices such that each color is used either [n/k] or [n/k] times. While classic Vertex Coloring is fixed parameter tractable under well established parameters such as pathwidth and feedback vertex set, equitable coloring is W[1]-hard. Little is known, however, about kernelization aspects of Equitable Coloring. In this work, we begin this investigation by first presenting a linear kernel for the parameter distance to clique, which contrasts with the quadratic kernel for Vertex Coloring under the same parameterization. Our main technical contribution is an OR-cross-composition from Multicolored Clique to Number List Coloring parameterized by vertex cover and number of colors which, along with two simple PPT reductions, implies that Equitable Coloring has no polynomial kernel under the same parameterization unless NP ⊆ coNP/poly.

In the Selective Coloring problem, we are given an integer k, a graph G, and a partition of V(G) into p parts, and the goal is to decide whether or not we can pick exactly one vertex of each part and obtain a k-colorable induced subgraph of G. This generalization of Vertex Coloring has only recently begun to be studied by Demange et al. [Theoretical Computer Science, 2014], motivated by scheduling problems on distributed systems, with Guo et al. [TAMC, 2020] discussing the first results on the parameterized complexity of the problem. In this work, we study multiple structural parameterizations for Selective Coloring. We begin by revisiting the many hardness results of Demange et al. and show how they may be used to provide intractability proofs for widely used parameters such as pathwidth, distance to co-cluster, and max leaf number. Afterwards, we present fixed parameter tractable algorithms when parameterizing by distance to cluster, or under the joint parameterizations treewidth and number of parts, and co-treewidth and number of colors. Our main contribution is a proof that, for every fixed k > 1, Selective Coloring does not admit a polynomial kernel when jointly parameterized by the vertex cover number and the number of parts, which implies that Multicolored Independent Set does not admit a polynomial kernel under the same parameterization.

For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ℋ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ℋ, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Given a graph G, a properk-coloring of G is a partition $$c = (S_i)_{i\in [1,k]}$$ of V(G) into k stable sets $$S_1,\ldots , S_{k}$$ . Given a weight function $$w: V(G) \rightarrow {\mathbb {R}}^+$$ , the weight of a color $$S_i$$ is defined as $$w(i) = \max _{v \in S_i} w(v)$$ and the weight of a coloringc as $$w(c) = \sum _{i=1}^{k}w(i)$$ . Guan and Zhu (Inf Process Lett 61(2):77–81, 1997) defined the weighted chromatic number of a pair (G, w), denoted by $$\sigma (G,w)$$ , as the minimum weight of a proper coloring of G. The problem of determining $$\sigma (G,w)$$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time $$n^{o(\log n)}$$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G, w) and an integer k, is $$\sigma (G,w) \le \sum _{v \in V(G)} w(v) - k$$ ? This parameterization has been recently considered by Escoffier (in: Proceedings of the 42nd international workshop on graph-theoretic concepts in computer science (WG). LNCS, vol 9941, pp 50–61, 2016), who provided an FPT algorithm running in time $$2^{{\mathcal {O}}(k \log k)} \cdot n^{{\mathcal {O}}(1)}$$ , and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time $$9^k \cdot n^{{\mathcal {O}}(1)}$$ , and prove that no algorithm in time $$2^{o(k)} \cdot n^{{\mathcal {O}}(1)}$$ exists under the ETH. On the other hand, we present a kernel with at most $$(2^{k-1}+1) (k-1)$$ vertices, and rule out the existence of polynomial kernels unless $$\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$$ , even on split graphs with only two different weights. Finally, we identify classes of graphs allowing for polynomial kernels, namely interval graphs, comparability graphs, and subclasses of circular-arc and split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.

In 2016, the first quantum processors have been made available to the general public. The possibility of programming an actual quantum device has elicited much enthusiasm. Yet, such possibility also brought challenges. One challenge is the so called Qubit Allocation problem: the mapping of a virtual quantum circuit into an actual quantum architecture. There exist solutions to this problem; however, in our opinion, they fail to capitalize on decades of improvements on graph theory. In contrast, this paper shows how to model qubit allocation as the combination of Subgraph Isomorphism and Token Swapping. This idea has been made possible by the publication of an approximative solution to the latter problem in 2016. We have compared our algorithm against five other qubit allocators, all independently designed in the last two years, including the winner of the IBM Challenge. When evaluated in "Tokyo", a quantum architecture with 20 qubits, our technique outperforms these state-of-the-art approaches in terms of the quality of the solutions that it finds and the amount of memory that it uses, while showing practical runtime.

In May of 2016, IBM Research has made a quantum processor available in the cloud to the general public. The possibility of programming an actual quantum device has elicited much enthusiasm. Yet, quantum programming still lacks the compiler support that modern programming languages enjoy today. To use universal quantum computers like IBM's, programmers must design low-level circuits. In particular, they must map logical qubits into physical qubits that need to obey connectivity constraints. This task resembles the early days of programming, in which software was built in machine languages. In this paper, we formally introduce the qubit allocation problem and provide an exact solution to it. This optimal algorithm deals with the simple quantum machinery available today; however, it cannot scale up to the more complex architectures scheduled to appear. Thus, we also provide a heuristic solution to qubit allocation, which is faster than the current solutions already implemented to deal with this problem.

Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i) = \max_{v \in S_i} w(v)$ and the weight of a coloring $c$ as $w(c) = \sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined the weighted chromatic number of a pair $(G,w)$, denoted by $\sigma(G,w)$, as the minimum weight of a proper coloring of $G$. The problem of determining $\sigma(G,w)$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on $n$-vertex trees in time $n^{o(\log n)}$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. In this article we provide some positive results for the problem, by considering its so-called dual parameterization: given a vertex-weighted graph $(G,w)$ and an integer $k$, the question is whether $\sigma(G,w) \leq \sum_{v \in V(G)} w(v) - k$. We prove that this problem is FPT by providing an algorithm running in time $9^k \cdot n^{O(1)}$, and it is easy to see that no algorithm in time $2^{o(k)} \cdot n^{O(1)}$ exists under the ETH. On the other hand, we present a kernel with at most $(2^{k-1}+1) (k-1)$ vertices, and we rule out the existence of polynomial kernels unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$, even on split graphs with only two different weights. Finally, we identify some classes of graphs on which the problem admits a polynomial kernel, in particular interval graphs and subclasses of split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.

In recent times, several activities that lately were considered possible to do only by human work, have been replaced by the use of artificial intelligence. Recent researches have been developed to give computers a creativity ability as main objective. A type of related research is algorithmic composition. Traditionally evolutionary algorithms are successfully used for composing tasks, using a wide variety of fitness functions. This work presents a multiobjective approach that combines Fux's rules, from music theory, and functions that capture melodies' proportions, based on the Zipf's law, to compose monophonic melodies. Besides that, we propose a method that measures the amount of creativity contained in an algorithmic composition method. Experimental results show that the proposed methods have a good musical quality.

In this work we tackle the bus stop selection step for the School Bus Routing Problem (SBRP). Our goal is to minimize the number of bus stops in order to assign all students to a bus stop respecting a home-to-bus-stop walking distance constraint. Our strategy creates a large number of possible bus stops points in a road network and uses a pseudo-random constructive heuristic algorithm to assign students to a bus stops. Our approach is tested on a real georeferenced data of a Brazilian city and is compared with a different methodology. Results demonstrate that the proposed approach is able to find good solutions for this optimization problem. Besides, the higher the number of possible points to install bus stops, the smaller is the number of bus stops required to attend all students.