Computer Science

Hashiwokakero, or simply Hashi, is a Japanese single-player puzzle played on a rectangular grid with no standard size. Some cells of the grid contain a circle, called island, with a number inside it ranging from one to eight. The remaining positions of the grid are empty. The player must connect all of the islands by drawing a series of horizontal or vertical bridges between them, respecting a series of rules: the number of bridges incident to an island equals the number indicated in the circle, at most two bridges are incident to any side of an island, bridges cannot cross each other or pass through islands, and each island must eventually be reachable from any other island. In this paper, we present some complexity results and relationships between Hashi and well-known graph theory problems. We give a formulation of the problem by means of an integer linear mathematical programming model, and apply a branch-and-cut algorithm to solve the model in which connectivity constraints are dynamically generated. We also develop a puzzle generator. Our experiments on 1440 Hashi puzzles show that the algorithm can consistently solve hard puzzles with up to 400 islands.

Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Best-of-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d= n α \,, where α=Ω((loglogn ) −1 ) . We prove that if initially each vertex is red with probability greater than 1/2+δ , and blue otherwise, where δ≥(logd ) −C for some C>0 , then with high probability this dynamic reaches a final state where all vertices are red within O(loglogn)+O(log( δ −1 )) steps.

In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced matching. We consider intermediate types of edge-colorings, where edges of some colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75) asserting that by allowing three additional induced matchings, one is able to save one matching color. We prove that every graph with maximum degree 3 can be decomposed into one matching and at most 8 induced matchings, and two matchings and at most 5 induced matchings. We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the above-mentioned conjecture for class I graphs.

A biclique of a graph is a maximal complete bipartite subgraph. The biclique graph of a graph G , KB(G) , defined as the intersection graph of the bicliques of G , was introduced and characterized in 2010. However, this characterization does not lead to polynomial time recognition algorithms. The time complexity of its recognition problem remains open. There are some works on this problem when restricted to some classes. In this work we give a characterization of the biclique graph of a K 3 -free graph G . We prove that KB(G) is the square graph of a particular graph which we call Mutually Included Biclique Graph of G ( K B m (G) ). Although it does not lead to a polynomial time recognition algorithm, it gives a new tool to prove properties of biclique graphs (restricted to K 3 -free graphs) using known properties of square graphs. For instance we generalize a property about induced P 3 ′ s in biclique graphs to a property about stars and proved a conjecture posted by Groshaus and Montero, when restricted to K 3 -free graphs. Also we characterize the class of biclique graphs of bipartite graphs. We prove that KB( bipartite )=( IIC-comparability ) 2 , where IIC-comparability is a subclass of comparability graphs that we call Interval Intersection Closed Comparability.

The Collatz graph is a directed graph with natural number nodes and where there is an edge from node x to node T(x)= T 0 (x)=x/2 if x is even, or to node T(x)= T 1 (x)= 3x+1 2 if x is odd. Studying the Collatz graph in binary reveals complex message passing behaviors based on carry propagation which seem to capture the essential dynamics and complexity of the Collatz process. We study the set E Pred k (x) that contains the binary expression of any ancestor y that reaches x with a limited budget of k applications of T 1 . The set E Pred k (x) is known to be regular, Shallit and Wilson [EATCS 1992]. In this paper, we find that the geometry of the Collatz graph naturally leads to the construction of a regular expression, reg k (x) , which defines E Pred k (x) . Our construction, is exponential in k which improves upon the doubly exponentially construction of Shallit and Wilson. Furthermore, our result generalises Colussi's work on the x=1 case [TCS 2011] to any natural number x , and gives mathematical and algorithmic tools for further exploration of the Collatz graph in binary.

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and (1− o n (1))⋅ 2 n 2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(nlogn) . Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered" is Eulerian.

Discrete modelling frameworks of Biological networks can be divided in two distinct categories: Boolean and Multi-valued. Although Multi-valued networks are more expressive for qualifying the regulatory behaviours modelled by more than two values, the ability to automatically convert them to Boolean network with an equivalent behaviour breaks down the fundamental borders between the two approaches. Theoretically investigating the conversion process provides relevant insights into bridging the gap between them. Basically, the conversion aims at finding a Boolean network bisimulating a Multi-valued one. In this article, we investigate the bisimilar conversion where the Boolean integer coding is a parameter that can be freely modified. Based on this analysis, we define a computational method automatically inferring a bisimilar Boolean network from a given Multi-valued one.

In this paper, we consider the following problem: given a connected graph G , can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP -hard when restricted to { P 6 , P 4 + P 2 } -free graphs and that it is coNP -hard when restricted to subcubic claw-free graphs and 2 P 3 -free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H -free graphs when H is connected.

In this paper, we study the problem of deciding whether the total domination number of a given graph G can be reduced using exactly one edge contraction (called 1-Edge Contraction( γ t )). We focus on several graph classes and determine the computational complexity of this problem. By putting together these results, we manage to obtain a complete dichotomy for H -free graphs.

A k -stack layout (also called a k -page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum k such that it admits a k -stack layout. A k -queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this decomposition result, we explore stack layouts of graph products. We show that the stack number is bounded for the strong product of a path and (i) a graph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree. The results are obtained via a novel concept of simultaneous stack-queue layouts, which may be of independent interest.