Aspects of distance measures in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G

Complexity and robustness of structures against extreme events

Civil structures are designed to support the loads acting on them. At present, the common practitioner considers both ordinary (winds, snow, accidental loads) and extreme events (earthquake, fire), combines the actions in such a way that, once the resistance of the elements is determined, the probability of failure is limited to a prescribed value. The set of events that may interest the structure is known and, therefore, a statistics of the actions is defined a priori. However, other events that are not forecastable may interest the construction. The sources of such events, called “Black Swans” after Taleb, are unknown, as well as their magnitude. For ensuring the integrity of the construction in such situations, which imply large damages, robust measures have to be taken (Chapter 3). Structural engineering is not the only domain in which unexpected events occur. Nature is the realm of contrasts. By means of evolution, living species differentiates, differentiated, in order to survive and reproduce. Various strategies were implemented in order to guarantee a biological robustness. Such mechanisms evoke one fundamental property of systems, the complexity and the connectivity between the components. The interaction between the parts makes the whole system more robust and tolerant to errors and damages (Chapters 1 and 2). Robustness in structures is implemented through classical strategies, which tend to limit the extent of damages through a design based on the consequences (Chapter 4). Being inspired by natural strategies, the idea of complexity in structural engineering is explored. Many issues arise, since a proper definition of this term has not been stated yet (Chapters 5 and 6). The ef- fects of element removal on frame structures, which represent an example of highly connected structural scheme, are investigated. As a result of simple simulations, the trend observed in Nature, which wants the complex systems to be robust to random damages, are spotted in the loaded structural schemes (Chapter 7)