Hitting minors, subdivisions, and immersions in tournaments

The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science. Difference with the previous version: use of the DMTCS article class. For a version with hyperlinks see the previous versio

On the complexity of acyclic modules in automata networks

Modules were introduced as an extension of Boolean automata networks. They have inputs which are used in the computation said modules perform, and can be used to wire modules with each other. In the present paper we extend this new formalism and study the specific case of acyclic modules. These modules prove to be well described in their limit behavior by functions called output functions. We provide other results that offer an upper bound on the number of attractors in an acyclic module when wired recursively into an automata network, alongside a diversity of complexity results around the difficulty of deciding the existence of cycles depending on the number of inputs and the size of said cycle.Comment: 21 page

Complexity of fixed point counting problems in Boolean Networks

A Boolean network (BN) with $n$ components is a discrete dynamical system described by the successive iterations of a function $f:\{0,1\}^n \to \{0,1\}^n$. This model finds applications in biology, where fixed points play a central role. For example, in genetic regulations, they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components: component $i$ has a positive (resp. negative) influence on component $j$ meaning that $j$ tends to mimic (resp. negate) $i$. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to a large number of BNs (which is, in average, doubly exponential according to $n$). The present work opens a new perspective on the well-established study of fixed points in BNs. When biologists discover the SID of a BN they do not know, they may ask: given that SID, can it correspond to a BN having at least/at most $k$ fixed points? Depending on the input, we prove that these problems are in $\textrm{P}$ or complete for $\textrm{NP}$, $\textrm{NP}^{\textrm{NP}}$, \textrm{NP}^{\textrm{#P}} or $\textrm{NEXPTIME}$. In particular, we prove that it is $\textrm{NP}$-complete (resp. $\textrm{NEXPTIME}$-complete) to decide if a given SID can correspond to a BN having at least two fixed points (resp. no fixed point).Comment: 43 page