Computer Science

We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges ( s can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and the cops win when they occupy all edges incident to the robbers position. We introduce the capture cost on G against a robber of speed s . This defines a hierarchy of invariants, namely δ 1 e , δ 2 e ,…, δ ∞ e , where δ ∞ e is an edge-analogue of the admissibility graph invariant, namely the {\em edge-admissibility} of a graph. We prove that the problem asking wether δ s e (G)≤k , is polynomially solvable when s∈{1,2,∞} while, otherwise, it is NP-complete. Our main result is a structural theorem for graphs of bounded edge-admissibility. We prove that every graph of edge-admissibility at most k can be constructed using (≤k) -edge-sums, starting from graphs whose all vertices, except possibly from one, have degree at most k . Our structural result is approximately tight in the sense that graphs generated by this construction always have edge-admissibility at most 2k−1 . Our proofs are based on a precise structural characterization of the graphs that do not contain θ r as an immersion, where θ r is the graph on two vertices and r parallel edges.

Many graph products have been applied to generate complex networks with striking properties observed in real-world systems. In this paper, we propose a simple generative model for simplicial networks by iteratively using edge corona product. We present a comprehensive analysis of the structural properties of the network model, including degree distribution, diameter, clustering coefficient, as well as distribution of clique sizes, obtaining explicit expressions for these relevant quantities, which agree with the behaviors found in diverse real networks. Moreover, we obtain exact expressions for all the eigenvalues and their associated multiplicities of the normalized Laplacian matrix, based on which we derive explicit formulas for mixing time, mean hitting time and the number of spanning trees. Thus, as previous models generated by other graph products, our model is also an exactly solvable one, whose structural properties can be analytically treated. More interestingly, the expressions for the spectra of our model are also exactly determined, which is sharp contrast to previous models whose spectra can only be given recursively at most. This advantage makes our model a good test-bed and an ideal substrate network for studying dynamical processes, especially those closely related to the spectra of normalized Laplacian matrix, in order to uncover the influences of simplicial structure on these processes.

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of (hyper)edges, that is, how, and how well, they are connected to others. In the case of directed networks, they assess the input they receive and the output they produce, and relations between them. With those tools, we can investigate biological networks. As examples, we apply our methods here to protein-protein interaction, transcriptional regulatory and metabolic networks.

For a simple graph G , a domination coloring of G is a proper vertex coloring such that every vertex of G dominates at least one color class, and every color class is dominated by at least one vertex. The domination chromatic number, denoted by χ dd (G) , is minimum number of colors among all domination colorings of G . In this paper, we discuss the effects of some typical operations on χ dd (G) , such as vertex (edge) removal, vertex (edge) contraction, edge subdivision, and cycle extending.

The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al.[AIEE Transactions 77 (1958)]. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length 2 n de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using O(n) time per bit and only O(n) space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.

In this paper, we are interested in algorithms that take in input an arbitrary graph G , and that enumerate in output all the (inclusion-wise) maximal "subgraphs" of G which fulfil a given property Π . All over this paper, we study several different properties Π , and the notion of subgraph under consideration (induced or not) will vary from a result to another. More precisely, we present efficient algorithms to list all maximal split subgraphs, sub-cographs and some subclasses of cographs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal independent sets of an auxiliary graph. For cographs and some subclasses , the algorithms rely on a framework recently introduced by Conte & Uno called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every "interesting" hereditary property Π . We extend the hardness result to some specific edge version of the extension problem.

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)⋅ n c for a computable function~ f and constant c whether the elimination distance of G to the class of degree d graphs is at most k .

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of d , the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least 9d−o(d) . It was shown recently that k≥2 robots traveling at unit speed also require at least 9d group search time. We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance x at constant speed s is given by s 2 x . Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple c of the distance d and the speed of the robots is bounded by b . Motivation for this study is that for the case when robots must complete the search in 9d time with maximum speed one, a single robot requires at least 9d energy, while for two robots, all previously proposed algorithms consume at least 28d/3 energy. When the robots have bounded memory, we generalize existing algorithms to obtain a family of optimal (and in some cases nearly optimal) algorithms parametrized by pairs of b,c values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. We also propose a novel search algorithm, with unbounded memory, that simultaneously achieves search time 9d and consumes energy 8.42588d . Our result shows that two robots can search on the line in optimal time 9d while consuming less total energy than a single robot within the same search time.

The main aim of decision support systems is to find solutions that satisfy user requirements. Often, this leads to predictability of those solutions, in the sense that having the input data and the model, an adversary or enemy can predict to a great extent the solution produced by your decision support system. Such predictability can be undesirable, for example, in military or security timetabling, or applications that require anonymity. In this paper, we discuss the notion of solution predictability and introduce potential mechanisms to intentionally avoid it.

Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide an output-polynomial time algorithm for incomparability graphs whose underlying posets have bounded dimension. Through a different proof technique, we also provide an output-polynomial algorithm for their complements, i.e., for comparability graphs of bounded dimension posets. Our algorithm for incomparability graphs is based on flashlight search and relies on the geometrical representation of incomparability graphs with bounded dimension, as given by Golumbic et al. in 1983. It runs with polynomial delay and only needs polynomial space. Our algorithm for comparability graphs is based on the flipping method introduced by Golovach et al. in 2015. It performs in incremental-polynomial time and requires exponential space. In addition, we show how to improve the flipping method so that it requires only polynomial space. Since the flipping method is a key tool for the best known algorithms enumerating minimal dominating sets in a number of graph classes, this yields direct improvements on the state of the art.