Rectangle Visibility Numbers of Graphs

Very-Large Scale Integration (VLSI) is the problem of arranging components on the surface of a circuit board and developing the wired network between components. One methodology in VLSI is to treat the entire network as a graph, where the components correspond to vertices and the wired connections correspond to edges. We say that a graph G has a rectangle visibility representation if we can assign each vertex of G to a unique axis-aligned rectangle in the plane such that two vertices u and v are adjacent if and only if there exists an unobstructed horizontal or vertical channel of finite width between the two rectangles that correspond to u and v. If G has such a representation, then we say that G is a rectangle visibility graph. Since it is likely that multiple components on a circuit board may represent the same electrical node, we may consider implementing this idea with rectangle visibility graphs. The rectangle visibility number of a graph G, denoted r(G), is the minimum k such that G has a rectangle visibility representation in which each vertex of G corresponds to at most k rectangles. In this thesis, we prove results on rectangle visibility numbers of trees, complete graphs, complete bipartite graphs, and (1,n)-hilly graphs, which are graphs where there is no path of length 1 between vertices of degree n or more

Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem

Evaluating Baseflow Recession Behavior using the Integrated Hydrologic Model ParFlow

Hydrologists have long studied the rate of streamflow recession as a means of understanding watershed properties. Historically, recession events were lumped together for analysis. However, more recent work has shown that individual recession events behave differently, and additional insights can be gained by evaluating events individually. The analysis of individual recession events has been shown to be a valuable tool for examining hydrologic processes in large watersheds, however its connection to hydrologic models has not been thoroughly explored. We used the integrated hydrologic model ParFlow to systematically explore the drivers of baseflow recession curves. Using ParFlow we demonstrated that the integrated model can generate shifting between recession events consistent with observational tudies. Furthermore, we found that storage has a major impact on recession curves, causing curves to shift to the right when storage is high. Subsurface configuration was also seen to have a large effect, with hydraulic conductivity influencing recession curves regardless of storage levels

Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is NP-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be NP-hard, which settles an open question.Peer Reviewe