Vadim E. Zverovich

The k-tuple domination number revisited

Abstract The following fundamental result for the domination number γ ( G ) of a graph G was proved by Alon and Spencer, Arnautov, Lovasz and Payan: γ ( G ) ≤ ln ( δ + 1 ) + 1 δ + 1 n , where n is the order and δ is the minimum degree of vertices of G . A similar upper bound for the double domination number was found by Harant and Henning [J. Harant, M.A. Henning, On double domination in graphs, Discuss. Math. Graph Theory 25 (2005) 29–34], and for the triple domination number by Rautenbach and Volkmann [D. Rautenbach, L. Volkmann, New bounds on the k -domination number and the k -tuple domination number, Appl. Math. Lett. 20 (2007) 98–102], who also posed the interesting conjecture on the k -tuple domination number: for any graph G with δ ≥ k − 1 , γ × k ( G ) ≤ ln ( δ − k + 2 ) + ln ( d k − 1 + d k − 2 ) + 1 δ − k + 2 n , where d m = ∑ i = 1 n ( d i m ) / n is the m -degree of G . This conjecture, if true, would generalize all the mentioned upper bounds and improve an upper bound proved in [A. Gagarin, V. Zverovich, A generalised upper bound for the k -tuple domination number, Discrete Math. (2007), in press ( doi:10.1016/j.disc.2007.07.033 )]. In this paper, we prove the Rautenbach–Volkmann conjecture.

On Roman, Global and Restrained Domination in Graphs

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed.

Randomized algorithms and upper bounds for multiple domination in graphs and networks

We consider four different types of multiple domination and provide new improved upper bounds for the k- and k-tuple domination numbers. They generalize two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets, whose expected orders satisfy new and recent upper bounds. The algorithms for k- and k-tuple dominating sets are of linear time in terms of the number of edges of the input graph, and they can be implemented as local distributed algorithms. Note that the corresponding multiple domination problems are known to be NP-complete.

Emergency Response in Complex Buildings: Automated Selection of Safest and Balanced Routes

The extreme importance of emergency response in complex buildings during natural and human-induced disasters has been widely acknowledged. In particular, there is a need for efficient algorithms for finding safest evacuation routes, which would take into account the 3-D structure of buildings, their relevant semantics, and the nature and shape of hazards. In this article, we propose algorithms for safest routes and balanced routes in buildings, where an extreme event with many epicenters is occurring. In a balanced route, a trade-off between route length and hazard proximity is made. The algorithms are based on a novel approach that integrates a multiattribute decision-making technique, Dijkstras classical algorithm and the introduced hazard proximity numbers, hazard propagation coefficient and proximity index for a route.

Upper bounds for the bondage number of graphs on topological surfaces

Abstract The bondage number b ( G ) of a graph G is the smallest number of edges of G whose removal results in a graph having the domination number larger than that of G . We show that, for a graph G having the maximum vertex degree Δ ( G ) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k , b ( G ) ≤ min { Δ ( G ) + h + 2 , Δ ( G ) + k + 1 } . This generalizes known upper bounds for planar and toroidal graphs, and can be improved for bigger values of the genera h and k by adjusting the proofs.

Discrepancy and signed domination in graphs and hypergraphs

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1s than -1s. This concept is closely related to combinatorial discrepancy theory as shown by Furedi and Mubayi [Z. Furedi, D. Mubayi, Signed domination in regular graphs and set-systems, J. Combin. Theory Ser. B 76 (1999) 223-239]. The signed domination number of G is the minimum of the sum of colours for all vertices, taken over all signed domination functions of G. In this paper, we present new upper and lower bounds for the signed domination number. These new bounds improve a number of known results.

The probabilistic approach to limited packings in graphs

We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k -limited packing if for every vertex v ? V ( G ) , | N v ? X | ? k , where N v is the closed neighbourhood of v . The k -limited packing number L k ( G ) of a graph G is the largest size of a k -limited packing in G . Limited packing problems can be considered as secure facility location problems in networks.In this paper, we develop a new application of the probabilistic method to limited packings in graphs, resulting in lower bounds for the k -limited packing number and a randomized algorithm to find k -limited packings satisfying the bounds. In particular, we prove that for any graph G of order n with maximum vertex degree Δ , L k ( G ) ? k n ( k + 1 ) ( Δ k ) ( Δ + 1 ) k . Also, some other upper and lower bounds for L k ( G ) are given.

Analytic Prioritization of Indoor Routes for Search and Rescue Operations in Hazardous Environments

Applications to prioritize indoor routes for emergency situations in a complex built facility have been restricted to building simulations and network approaches. These types of applications often failed to account for the complexity and trade-offs needed to select the optimal indoor path during an emergency situation. In this article, we propose a step change for finding the optimal routes for Search And Rescue (SAR) teams in a building, where a multi-epicentre extreme event is occurring. We have developed an algorithm that is based on a novel approach integrating the Analytic Hierarchy Process (AHP), statistical characteristics, the propagation of hazard, Duckham-Kulik’s adapted algorithm, Dijkstras classical algorithm, and the binary search with three criteria: hazard proximity, distance/travel time, and route complexity. The sub-criteria for the route complexity are validated in the context of SAR using a real-life building (Doha World Trade Centre). The important feature of the algorithm is its ability to generate an optimal route depending on user’s needs. The findings revealed that the generated optimal routes are indeed the ‘best’ trade-off amongst distance/travel time, hazard proximity and route complexity. The test results also demonstrated the robustness of the algorithm with respect to different parameters, and its insensitivity to different scenarios of uncontrolled evacuation.

3D Capture Techniques for BIM Enabled LCM

As a special kind of Product Life cyle Management (PLM), Building Life cycle Management (BLM) is a centric activity for facility owners and managers. This fact motivates the adoption of Building Information Modeling (BIM) approaches as a way to achieve smart BLM strategies for cost reduction, facility knowledge management, and project synchronization among the different stakeholders. Unfortunately, the current BIM state of the art is tailored towards the management of new projects, while ongoing and completed AEC projects could hugely benefit from BIM integration for better BLM strategies. In this regards, it is absolutely necessary to acquire knowledge about the dynamic facility aspects (crowd movement, as-is updates, etc.). Up-to-date, 3D capture appears to be the only reliable way to cope with such situation. In this paper, we analyze 3D capture techniques, ranging from photogrammetry to 3D scanning, with an emphasis on helping 3D capture practitioners to make critical decisions about the choice of adequate acquisition technologies for a particular application. We discuss 3D capture techniques by exposing their pros and cons, according to several relevant criteria, and synthesize our analysis by developing a set of recommendations to enhance the life expectancy of buildings via the integration of BIM into Life Cycle Management (LCM) of the built environment and its buildings.

On general frameworks and threshold functions for multiple domination

We consider two general frameworks for multiple domination, which are called -domination and parametric domination. They generalise and unify -domination, k -domination, total k -domination and k -tuple domination. In this paper, known upper bounds for the classical domination are generalised for the -domination and parametric domination numbers. These generalisations are based on the probabilistic method and they imply new upper bounds for the -domination and total k -domination numbers. Also, we study threshold functions, which impose additional restrictions on the minimum vertex degree, and present new upper bounds for the aforementioned numbers. Those bounds extend similar known results for k -tuple domination and total k -domination.