On the Connected Safe Number of Some Classes of Graphs

For a connected simple graph G , a nonempty subset S of V G is a connected safe set if the induced subgraph G S is connected and the inequality S ≥ D satisfies for each connected component D of G∖S whenever an edge of G exists between S and D . A connected safe set of a connected graph G with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.

Symbolic Regression for Approximating Graph Geodetic Number

Graph properties are certain attributes that could make the structure of the graph understandable. Occasionally, standard methods cannot work properly for calculating exact values of graph properties due to their huge computational complexity, especially for real-world graphs. In contrast, heuristics and metaheuristics are alternatives proved their ability to provide sufficient solutions in a reasonable time. Although in some cases, even heuristics are not efficient enough, where they need some not easily obtainable global information of the graph. The problem thus should be dealt in completely different way by trying to find features that related to the property and based on these data build a formula which can approximate the graph property. In this work, symbolic regression with an evolutionary algorithm called Cartesian Genetic Programming has been used to derive formulas capable to approximate the graph geodetic number which measures the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Finding the exact value of the geodetic number is known to be NP-hard for general graphs. The obtained formulas are tested on random and real-world graphs. It is demonstrated how various graph properties as training data can lead to diverse formulas with different accuracy. It is also investigated which training data are really related to each property.

Minimal Cardinality Diagnosis in Problems with Multiple Observations

Model-Based Diagnosis (MBD) is a well-known approach to diagnosis in medical domains. In this approach, the behavior of a system is modeled and used to identify faulty components, i.e., once a symptom of abnormal behavior is observed, an inference algorithm is run on the system model and returns possible explanations. Such explanations are referred to as diagnoses. A diagnosis is an assumption about which set of components are faulty and have caused the abnormal behavior. In this work, we focus on the case where multiple observations are available to the diagnoser, collected at different times, such that some of these observations exhibit symptoms of abnormal behavior. MBD with multiple observations is challenging because some components may fail intermittently, i.e., behave abnormally in one observation and behave normally in another, while other components may fail all the time (non-intermittently). Inspired by recent success in solving classical diagnosis problems using Boolean satisfiability (SAT) solvers, we describe two SAT-based approaches to solve this MBD with multiple observations problem. The first approach compiles the problem to a single SAT formula, and the second approach solves each observation independently and then merges them together. We compare these two approaches experimentally on a standard diagnosis benchmark and analyze their pros and cons.

Metric Dimension of Generalized Möbius Ladder and its Application to WSN Localization

Localization is one of the key techniques in wireless sensor network. While the global positioning system (GPS) is one of the most popular positioning technologies, the weakness of high cost and energy consuming makes it difficult to install in every node. In order to reduce the cost and energy consumption only a few nodes, called beacon nodes, are equipped with GPS modules. The remaining nodes obtain their locations through localization. In order to find the minimum positions of beacons, a resolving set with minimal cardinality has been obtained in the network which is called metric basis. Simultaneous local metric basis of the network is also given in which each pair of adjacent vertices of the network is distinguished by some element of simultaneous local metric basis which makes the network design more reasonable. In this paper a new network, the generalized Möbius ladder Mm,n, has been introduced and its metric dimension and simultaneous local metric dimension of its two subfamilies have been calculated.

Cyclic covering of a module over an Artinian ring

Given a commutative ring with identity [Formula: see text] and an [Formula: see text]-module [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a cyclic covering of [Formula: see text], if this module is the union of the cyclic submodules [Formula: see text], where [Formula: see text]. Such covering is said to be irredundant, if no proper subset of [Formula: see text] is a cyclic covering of [Formula: see text]. In this work, an irredundant cyclic covering of [Formula: see text] is constructed for every Artinian commutative ring [Formula: see text]. As a consequence, a cyclic covering of minimal cardinality of [Formula: see text] is obtained for every finite commutative ring [Formula: see text], extending previous results in the literature.

Essential circles and Gromov–Hausdorff convergence of covers

The [Formula: see text]-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of covering maps of compact geodesic spaces called “circle covers” that are “closed” with respect to Gromov–Hausdorff convergence and include [Formula: see text]-covers. In fact, we use circle covers to completely understand the limiting behavior of [Formula: see text]-covers. The proofs use the descrete homotopy methods developed by Berestovskii, Plaut, and Wilkins, and in fact we show that when [Formula: see text], the Sormani–Wei [Formula: see text]-cover is isometric to the Berestovskii–Plaut–Wilkins [Formula: see text]-cover. Of possible independent interest, our arguments involve showing that “almost isometries” between compact geodesic spaces result in explicitly controlled quasi-isometries between their [Formula: see text]-covers. Finally, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.

Choice of the control variables of an isolated intersection by graph colouring

This paper deals with the problem of grouping the traffic streams into some groups - signal groups on a signalized intersection. The fact that more traffic streams, which are not in a conflict, can be controlled by one sequence of traffic lights means that one control variable can be assigned to one signal group. Determination of the complete sets of signal groups, i.e. the groups of traffic streams on one intersection, controlled by one control variable is defined in this paper as a graphcoloring problem. The complete sets of signal groups are obtained by coloring the complement of the graph of identical indications. It is shown that the minimal number of signal groups in the complete set of signal groups is equal to the chromatic number of the complement of the graph with identical indications. The problem of finding all complete sets of signal groups with minimal cardinality, which is equal to the chromatic number, is formulated as a linear programming problem where the values of variables belong to set {0,1}.

A Novel SAT-Based Approach to Model Based Diagnosis

This paper introduces a novel encoding of Model Based Diagnosis (MBD) to Boolean Satisfaction (SAT) focusing on minimal cardinality diagnosis. The encoding is based on a combination of sophisticated MBD preprocessing algorithms and the application of a SAT compiler which optimizes the encoding to provide more succinct CNF representations than obtained with previous works. Experimental evidence indicates that our approach is superior to all published algorithms for minimal cardinality MBD. In particular, we can determine, for the first time, minimal cardinality diagnoses for the entire standard ISCAS-85 and 74XXX benchmarks. Our results open the way to improve the state-of-the-art on a range of similar MBD problems.