Identifying and mitigating the spread of fake information is a challenging issue that has become dominant with the rise of social media. We consider a generalization of the Domination problem that can be used to detect a set of individuals who, once immunized, can prevent the spreading of fake narratives. The considered problem, named Distance Vector Domination generalizes both distance and multiple domination, at individual (i.e., vertex) level. We study the parameterized complexity of the problem according to several standard and structural parameters. We prove the W[1]-hardness of the problem with respect to neighborhood diversity, even when all the distances are 1. We also give fixed-parameter algorithms for some variants of the problem and parameter combinations.

In this paper, we provide an efficient algorithm to construct almost optimal (k,n,d)-superimposed codes with runlength constraints. A (k,n,d)-superimposed code of length t is a t×n binary matrix such that any two 1’s in each column are separated by a run of at least d 0’s, and such that for any column c and any other k−1 columns, there exists a row where c has 1 and all the remaining k−1 columns have 0. These combinatorial structures were introduced by Agarwal et al. (2020), in the context of Non-Adaptive Group Testing algorithms with runlength constraints.By using Moser and Tardos’ constructive version of the Lovász Local Lemma, we provide an efficient randomized Las Vegas algorithm of complexity Θ(tn2) for the construction of (k,n,d)-superimposed codes of length t=O(dklogn+k2logn). We also show that the length of our codes is shorter, for n sufficiently large, than that of the codes whose existence was proved in Agarwal et al. (2020).

The Identifying Code (IC) problem seeks a vertex subset whose intersection with every vertex's closed neighborhood is unique, enabling fault detection in multiprocessor systems and practical uses in identity verification, environmental monitoring, and dynamic localization. A closely related problem is the Locating-Dominating Set (LD), which requires each non-dominating vertex to be uniquely identified by its intersection with the set. Cappelle, Gomes, and Santos (2021) proved that LD is W-hard for minimum clique cover and lacks polynomial kernels for parameters such as vertex cover, but their methods did not apply to IC. This paper answers their question by showing that IC does not admit a polynomial kernel parameterized by solution size plus vertex cover unless NP is a subset of coNP/poly.

With the rise of social media, misinformation has significant negative effects on the decision-making of individuals, organizations and communities within society. Identifying and mitigating the spread of fake information is a challenging issue. We consider a generalization of the Domination problem that can be used to detect a set of individuals who, through an awareness process, can prevent the spreading of fake narratives. The considered problem, named (d,r)-Domination generalizes both distance and multiple domination. We study the parameterized complexity of the problem according to standard and structural parameters. We give fixed-parameter algorithms as well as polynomial compressions/kernelizations for some variants of the problem and parameter combinations.

The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree T in a given graph G with the fewest branch vertices, defined as vertices with a degree three or more in T. This problem, known to be NP-hard, has attracted significant attention due to its importance in network design and optimization. Extensive research has been conducted on the algorithmic and combinatorial aspects of this problem, with recent studies delving into its fixed-parameter tractability. In this paper, we focus primarily on the parameter modular-width. We demonstrate that finding a spanning tree with the minimum number of branch vertices is Fixed-Parameter Tractable (FPT) when considered with respect to modular-width. Additionally, in cases where each vertex in the input graph has an associated cost for serving as a branch vertex, we prove that the problem of finding a spanning tree with the minimum branch cost (i.e., minimizing the sum of the costs of branch vertices) is FPT with respect to neighborhood diversity.

This Special Issue is dedicated to a selection of papers from the 35th International Workshop on Combinatorial Algorithms (IWOCA 2024), held in July 2024, in Ischia, Italy, with Adele A [...]

We introduce the Red-Blue Unshared Domination (RBUD) problem, a variant of domination problems in graphs. Given an undirected graph G with vertices colored either red or blue, the goal is to find a minimum-size subset of vertices S such that for every red-blue pair of vertices outside S, there exist two distinct vertices in S (called unshared dominators), each dominating one vertex of the pair but not the other. Specifically, for each red vertex r ∉S and blue vertex b∉S , there must be a vertex in S that dominates r but not b and another vertex in S that dominates b but not r. We prove that the RBUD problem is NP-hard, even for bipartite planar subcubic graphs with large girth. We also provide an approximation algorithm for general graphs and exact efficient algorithms for trees and graphs with bounded neighborhood diversity. Additionally, we extend our algorithmic approach to graphs with bounded modular-width.

This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity.Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique, Minimum Graph Coloring, (Total) Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above.

Parameterized complexity, introduced to efficiently solve NP-hard problems for small values of a fixed parameter, has been recently used as a tool to speed up algorithms for tractable problems. Following this line of research, we design algorithms parameterized by neighborhood diversity (nd$$ \mathsf{nd} $$) for several graph theoretic problems in P$$ P $$: Maximum b$$ b $$-Matching, Triangle Counting and Listing, Girth, Global Minimum Vertex Cut, and Perfect Graphs Recognition. Such problems are known to admit algorithms parameterized by modular-width (mw$$ \mathsf{mw} $$) and consequently-as nd$$ \mathsf{nd} $$ is a special case of mw$$ \mathsf{mw} $$-by nd$$ \mathsf{nd} $$. However, the proposed novel algorithms allow for improving the computational complexity from time O(f(mw)n+m)$$ O\left(f\left(\mathsf{mw}\right)\cdotp n+m\right) $$-where n$$ n $$ and m$$ m $$ denote, respectively, the number of vertices and edges in the input graph-to time O(g(nd)+n+m)$$ O\left(g\left(\mathsf{nd}\right)+n+m\right) $$ which is only additive in the size of the input. Then we consider some classical NP-hard problems (Maximum independent set, Maximum clique, and Minimum dominating set) and show that for several classes of hereditary graphs, they admit linear time algorithms for sufficiently small-nonnecessarily constant-values of the neighborhood diversity parameter.

List union-free families are basic combinatorial structures that appear in different application scenarios, most notably in one-bit compressed sensing. In this paper, we study algorithms for the construction of list union-free families and we provide bounds on the parameters of these families that substantially affect the complexity of the algorithms that utilize them.

Graph burning is a deterministic, discrete-time process that can be used to model how influence or contagion spreads in a graph. In the graph burning process, each node starts as dormant, and becomes informed/burned over time; when a node is burned, it remains burned until the end of the process. In each round, one can burn a new node (source of fire) in the network. Once a node is burned in round $t$, in round $t+1$, each of its dormant neighbors becomes burned. The process ends when all nodes are burned; the goal is to minimize the number of rounds.We study a variation of graph burning in order to model spreading processes in community-based networks. With respect to a specific piece of information, a community is {\em satisfied} when this information reaches at least a prescribed number of its members. Specifically, we consider the problem of identifying a minimum length sequence of nodes that, according to a graph burning process, allows to satisfy all the communities of the network. We investigate this NP-hard problem from an approximation point of view, showing both a lower bound and a matching upper bound. We also investigate the case when the number of communities is constant and show how to solve the problem with a constant approximation factor.Moreover, we consider the problem of maximizing the number of satisfied groups, given a budget $k$ on the number of rounds.

We introduce a class of generalized superimposed codes that include several cases already studied in the literature. We give bounds on their size and algorithms for their construction.

In this paper, we study upper bounds on the minimum length of frameproof codes introduced by Boneh and Shaw [3] to protect copyrighted materials. A q-ary (k,n)-frameproof code of length t is a t×n matrix having entries in {0,1,…,q−1} and with the property that for any column c and any other k columns, there exists a row where the symbols of the k columns are all different from the corresponding symbol (in the same row) of the column c. In this paper, we show the existence of q-ary (k,n)-frameproof codes of length $t = O\left( {\frac{{{k^2}}}{q}\log n} \right)$ for q ≤ k, using the Lovász Local Lemma, and of length $t = O\left( {\frac{{{k^2}}}{{\log \left( {q/k} \right)}}\log \left( {n/k} \right)} \right)$ for q > k using the expurgation method. Remarkably, for the practical case of q ≤ k our findings give codes whose length almost matches the lower bound $\Omega \left( {\frac{{{k^2}}}{{q\log k\log n}}} \right)$ on the length of any q-ary (k,n)-frameproof code and, more importantly, allow us to derive an algorithm of complexity O(tn 2 ) for the construction of such codes.

A branch vertex in a tree is a vertex of degree at least three. We study the NP-hard problem of constructing spanning trees with as few branch vertices as possible. This problem generalizes the famous Hamiltonian Path problem which corresponds to the case of no vertices having degree three or more. It has been extensively studied in the literature and has important applications in network design and optimization. In this paper, we study the problem of finding a spanning tree with the minimum number of branch vertices in graphs of bounded neighborhood diversity. Neighborhood diversity, a generalization of vertex cover to dense graphs, plays an important role in the design of algorithms for such graphs.

Graph burning is a deterministic, discrete-time process that can be used to model how influence or contagion spreads in a graph. In the graph burning process, each node starts as dormant, and becomes informed/burned over time; when a node is burned, it remains burned until the end of the process. In each round, one can burn a new node (source of fire) in the network. Once a node is burned in round t, in round $$t+1$$ , each of its dormant neighbors becomes burned. The process ends when all nodes are burned; the goal is to minimize the number of rounds. We study a variation of graph burning in order to model spreading processes in community-based networks. With respect to a specific piece of information, a community is satisfied when this information reaches at least a prescribed number of its members. Specifically, we consider the problem of identifying a minimum length sequence of nodes that, according to a graph burning process, allows to satisfy all the communities of the network. We investigate this NP-hard problem from an approximation point of view, showing both a lower bound and a matching upper bound. We also investigate the case when the number of communities is constant and show how to solve the problem with a constant approximation factor. Moreover, we consider the problem of maximizing the number of satisfied groups, given a budget k on the number of rounds.

The minimum branch vertices spanning tree problem consists in finding a spanning tree T of an input graph G having the minimum number of branch vertices, that is, vertices of degree at least three in T . This NP -hard problem has been widely studied in the literature and has many important applications in network design and optimization. Algorithmic and combinatorial aspects of the problem have been extensively studied and its fixed parameter tractability has been recently considered. In this paper we focus on modular-width and show that the problem of finding a spanning tree with the minimum number of branch vertices is FPT with respect to this parameter.