Computer Science

Automata networks are a very general model of interacting entities, with applications to biological phenomena such as gene regulation. In many contexts, the order in which entities update their state is unknown, and the dynamics may be very sensitive to changes in this schedule of updates. Since the works of Aracena et. al, it is known that update digraphs are pertinent objects to study non-equivalent block-sequential update schedules. We prove that counting the number of equivalence classes, that is a tight upper bound on the synchronism sensitivity of a given network, is #P-complete. The problem is nevertheless computable in quasi-quadratic time for oriented cacti, and for oriented series-parallel graphs thanks to a decomposition method.

In a graph G=(V,E) , a module is a vertex subset M of V such that every vertex outside M is adjacent to all or none of M . For example, ??, {x} (x?�V) and V are modules of G , called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex x of a prime graph G is critical if G?�x is decomposable. Moreover, a prime graph with k non-critical vertices is called (?�k) -critical graph. A prime graph G is k -minimal if there is some k -vertex set X of vertices such that there is no proper induced subgraph of G containing X is prime. From this perspective, I. Boudabbous proposes to find the (?�k) -critical graphs and k -minimal graphs for some integer k even in a particular case of graphs. This research paper attempts to answer I. Boudabbous's question. First, it describes the (?�k) -critical tree. As a corollary, we determine the number of nonisomorphic (?�k) -critical tree with n vertices where k?�{1,2,??n 2 ?�} . Second, it provide a complete characterization of the k -minimal tree. As a corollary, we determine the number of nonisomorphic k -minimal tree with n vertices where k?? .

We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in terms of their induced subgraphs (while such a characterization is well-understood in terms of minors). They also provide examples of graphs of large treewidth and large rankwidth in well-studied classes, such as (theta, triangle)-free graphs and even-hole-free graphs with no K 4 (where a hole is a chordless cycle of length at least four, a theta is a graph made of three internally vertex disjoint paths of length at least two linking two vertices, and K 4 is the complete graph on four vertices).

A {\em theta} is a graph made of three internally vertex-disjoint chordless paths P 1 =a…b , P 2 =a…b , P 3 =a…b of length at least~2 and such that no edges exist between the paths except the three edges incident to a and the three edges incident to b . A {\em pyramid} is a graph made of three chordless paths P 1 =a… b 1 , P 2 =a… b 2 , P 3 =a… b 3 of length at least~1, two of which have length at least 2, vertex-disjoint except at a , and such that b 1 b 2 b 3 is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to~ a . An \emph{even hole} is a chordless cycle of even length. For three non-negative integers i≤j≤k , let S i,j,k be the tree with a vertex v , from which start three paths with i , j , and k edges respectively. We denote by K t the complete graph on t vertices. We prove that for all non-negative integers i,j,k , the class of graphs that contain no theta, no K 3 , and no S i,j,k as induced subgraphs have bounded treewidth. We prove that for all non-negative integers i,j,k,t , the class of graphs that contain no even hole, no pyramid, no K t , and no S i,j,k as induced subgraphs have bounded treewidth. To bound the treewidth, we prove that every graph of large treewidth must contain a large clique or a minimal separator of large cardinality.

Modular Decomposition focuses on repeatedly identifying a module M (a collection of vertices that shares exactly the same neighbourhood outside of M) and collapsing it into a single vertex. This notion of exactitude of neighbourhood is very strict, especially when dealing with real world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an ({\alpha}, {\beta})-module, a relaxation that allows a bounded number of errors in each node and maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal ({\alpha}, {\beta})-modules can be computed in polynomial time, and that every graph admits an ({\alpha},{\beta})-modular decomposition tree, thus generalizing Gallai's Theorem (which corresponds to the case for {\alpha} = {\beta} = 0). Unfortunately we give evidence that computing such a decomposition tree can be difficult.

The 2 -layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of k -planar graphs has been considered only for k=1 in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of 2 -layer k -planar graphs with k∈{2,3,4,5} . Based on these results, we provide a Crossing Lemma for 2 -layer k -planar graphs, which then implies a general density bound for 2 -layer k -planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between k -planarity and h -quasiplanarity in the 2 -layer model and show that 2 -layer k -planar graphs have pathwidth at most k+1 .

In the Many-visits Path TSP, we are given a set of n cities along with their pairwise distances (or cost) c(uv) , and moreover each city v comes with an associated positive integer request r(v) . The goal is to find a minimum-cost path, starting at city s and ending at city t , that visits each city v exactly r(v) times. We present a 3 2 -approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in n and poly-logarithmic in the requests r(v) . Our algorithm can be seen as a far-reaching generalization of the 3 2 -approximation algorithm for Path TSP by Zenklusen (SODA 2019), which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP. One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree bounded multigraph of minimum cost. We tackle this problem by generalizing a fundamental result of Király, Lau and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for general polymatroids, even allowing element multiplicities. Our result directly yields a 3 2 -approximation to the metric Many-visits TSP, as well as a 3 2 -approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.

Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a category should have. The computation is modelled by a functor that encodes updating the computation, and we give an abstract account of such functors. We show that every updating functor satisfying our conditions is computable.

A class of models intended to be as minimal and structureless as possible is introduced. Even in cases with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some important core known features of fundamental physics are seen, suggesting the possibility that the models may provide a new approach to finding a fundamental theory of physics.

The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k ≥ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph.