We used a divide-and-conquer algorithm to recursively solve the two-dimensional problem of protein folding of an HP sequence with the maximum number of H-H contacts. We derived both lower and upper bounds for the algorithmic complexity by using the newly introduced concept of multi-directional width-bounded geometric separator. We proved that for a grid graph G with n grid points P, there exists a balanced separator A subseteq P$ such that A has less than or equal to 1.02074 sqrt{n} points, and G-A has two disconnected subgraphs with less than or equal to {2over 3}n nodes on each subgraph. We also derive a 0.7555sqrt {n} lower bound for our balanced separator. Based on our multidirectional width-bounded geometric separator, we found that there is an O(n^{5.563sqrt{n}}) time algorithm for the 2D protein folding problem in the HP model. We also extended the upper bound results to rectangular and triangular lattices.

The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides a characterization of planar graphs in terms of their poset dimension, as follows: a graph G is planar if and only if the dimension of the incidence poset of G is at most three. One of the implications of the theorem is proved by giving an explicit mapping of the vertices to R^2 that defines a straightline embedding of the graph. The other implication is proved by introducing the concept of normal labelling. Normal labellings of plane triangulations can be used to obtain a realizer of the incidence poset. We present an exposition of Schnyder’s theorem with his original proof, using normal labellings. An alternate proof of Schnyder’s Theorem is also presented. This alternate proof does not use normal labellings, instead we use some structural properties of a realizer of the incidence poset to deduce the result. Some applications and a generalization of one implication of Schnyder’s Theorem are also presented in this work. Normal labellings of plane triangulations can be used to obtain a barycentric embedding of a plane triangulation, and they also induce a partition of the edge set of a plane triangulation into edge disjoint trees. These two applications of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and canonical orderings to obtain a compact drawing of a graph, are also presented. A generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s Theorem was proved by Ossona de Mendez. This generalization is also presented in this work. The concept of order labelling is also introduced and we show some similarities of the order labelling and the normal labelling. Finally, we conclude this work by showing the source code of some implementations done in Sage.

The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides a characterization of planar graphs in terms of their poset dimension, as follows: a graph G is planar if and only if the dimension of the incidence poset of G is at most three. One of the implications of the theorem is proved by giving an explicit mapping of the vertices to R^2 that defines a straightline embedding of the graph. The other implication is proved by introducing the concept of normal labelling. Normal labellings of plane triangulations can be used to obtain a realizer of the incidence poset. We present an exposition of Schnyder’s theorem with his original proof, using normal labellings. An alternate proof of Schnyder’s Theorem is also presented. This alternate proof does not use normal labellings, instead we use some structural properties of a realizer of the incidence poset to deduce the result. Some applications and a generalization of one implication of Schnyder’s Theorem are also presented in this work. Normal labellings of plane triangulations can be used to obtain a barycentric embedding of a plane triangulation, and they also induce a partition of the edge set of a plane triangulation into edge disjoint trees. These two applications of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and canonical orderings to obtain a compact drawing of a graph, are also presented. A generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s Theorem was proved by Ossona de Mendez. This generalization is also presented in this work. The concept of order labelling is also introduced and we show some similarities of the order labelling and the normal labelling. Finally, we conclude this work by showing the source code of some implementations done in Sage.

We study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2...n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses thereof. In Chapter 1 we introduce the topic of this thesis and in Chapter 2 we review relevant definitions and basic results. In Chapter 3 we prove that the maximum differential coloring problem remains NP-hard even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs and close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs. In Chapter 4 we introduce the (d, kn)-differential coloring problem. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, in this chapter we provide a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ([2n/3], 2n)-differential coloring problem, even when input graph is planar. In Chapter 5 we experimentally evaluate and compare several algorithms for coloring a map. Motivated by different application scenarios, we classify our approaches into two categories, depending on the dimensionality of the underlying color space. To cope with the one dimensional color space (e.g., gray-scale colors), we employ the (d, kn)-differential coloring. In Chapter 6 we describe a practical approach for visualizing multiple relationships defined on the same dataset using a geographic map metaphor, where clusters of nodes form countries and neighboring countries correspond to nearby clusters. The aim is to provide a visualization that allows us to compare two or more such maps. In the case where we are considering multiple relationships we also provide an interactive tool to visually explore the effect of combining two or more such relationships. Our method ensures good readability and mental map preservation, based on dynamic node placement with node stability, dynamic clustering with cluster stability, and dynamic coloring with color stability. Finally in Chapter 7 we discuss future work and open problems.

For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while a few knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and attached metal structures. In this thesis, we survey the known types of knots, links, and non-planar graphs in protein structures with and without including such bonds between proteins and metals. Then we present new examples of protein structures containing M\"obius ladders and other non-planar graphs as a result of these bound metal atoms. Finally, we propose hypothetical structures illustrating specific disulfide connectivities that would result in the key ring link, the Whitehead link and the 5_1 knot, the latter two of which have thus far not been identified within protein structures.

A labeled embedding of a planar graph G is a pair (G, g) consisting of a planar drawing G of G and a function g assigning labels (colors) to the faces of G. We study the problem of Embedding Restriction Satisfiability (ERS) that investi- gates whether a given graph has a labeled embedding satisfying a provided set of conditions. ERS is a relatively new problem, so not much is known about it. Nevertheless, it has great potential. It generalizes several problems looking for a particular drawing of a planar graph, such as the problem of Partially Embedded Planarity. Therefore, ERS may become a focal point in the area of graph draw- ing. In this thesis, we examine the computational complexity of ERS. We show that ERS is NP-complete. After that, we look at the complexity of some specific classes of its instances. We try to locate the boundary between the NP-complete and the polynomial variants of the problem. 1

A graph Gis said to be an intersection graph of sets from a family of sets if there exists a function ƒ : V(G)→ such that for u,v V(G), (u,v) E(G) ƒ (u) ƒ (v) ≠ . Interval graphs are thus the intersection graphs of closed intervals on the real line and unit interval graphs are the intersection graphs of unit length intervals on the real line. An interval on the real line can be generalized to a “kbox” in Rk.A kbox B =(R1,R2,...,Rk), where each Riis a closed interval on the real line, is defined to be the Cartesian product R1x R2x…x Rk. If each Ri is a unit length interval, we call B a k-cube. Thus, 1-boxes are just closed intervals on the real line whereas 2-boxes are axis-parallel rectangles in the plane. We study the intersection graphs of k-boxes and k-cubes. The parameter boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. Thus, interval graphs are the graphs with boxicity at most 1 and unit interval graphs are the graphs with cubicity at most 1. These parameters were introduced by F. S.Roberts in 1969. In some sense, the boxicity of a graph is a measure of how different a graph is from an interval graph and in a similar way, the cubicity is a measure of how different the graph is from a unit interval graph. We prove several upper bounds on the boxicity and cubicity of general as well as special classes of graphs in terms of various graph parameters such as the maximum degree, the number of vertices and the bandwidth. The following are some of the main results presented. 1. We show that for any graph G with maximum degree Δ , box(G)≤ Δ 22 . This result implies that bounded degree graphs have bounded boxicity no matter how large the graph might be. 2. It was shown in [18] that the boxicity of a graph on n vertices with maximum degree Δ is O(Δ ln n). But a similar bound does not hold for the average degree davof a graph. [18] gives graphs in which the boxicity is exponentially larger than davln n. We show that even though an O(davln n) upper bound for boxicity does not hold for all graphs, for almost all graphs, boxicity is O(davln n). 3. The ratio of the cubicity to boxicity of any graph shown in [15] when combined with the results on boxicity show that cub(G) is O(Δ ln 2 n) and O(2 ln n) for any graph G on n vertices and with maximum degree . By using a randomized construction, we prove the better upper bound cub(G) ≤ [4(Δ + 1) ln n.] 4. Two results relating the cubicity of a graph to its bandwidth b are presented. First, it is shown that cub(G) ≤ 12(Δ + 1)[ ln(2b)] + 1. Next, we derive the upper bound cub(G) ≤ b + 1. This bound is used to derive new upper bounds on the cubicity of special graph classes like circular arc graphs, cocomparability graphs and ATfree graphs in relation to the maximum degree. 5. The upper bound for cubicity in terms of the bandwidth gives an upper bound of Δ + 1 for the cubicity of interval graphs. This bound is improved to show that for any interval graph G with maximum degree , cub(G) ≤[ log2 Δ] + 4. 6. Scheinerman [54] proved that the boxicity of any outerplanar graph is at most 2. We present an independent proof for the same theorem. 7. Halin graphs are planar graphs formed by adding a cycle connecting the leaves of a tree none of whose vertices have degree 2. We prove that the boxicity of any Halin graph is equal to 2 unless it is a complete graph on 4 vertices, in which case its boxicity is 1.

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where $W=\{u,v\}$ and $G$ has a separating $C_3$ or $C_4$ between $u$ and $v$.

This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees. In the Propagation problem we are given a graph $G$ and the goal is to find a minimum-sized set of nodes $S$ that covers all of the nodes, where a node $v$ is covered if (1) $v$ is in $S$, or (2) $v$ has a neighbor $u$ such that $u$ and all of its neighbors except $v$ are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use $n$ to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in weighted (general) graphs. The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem. We show that it is hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in general graphs. We design and analyze an approximation algorithm with a performance guarantee of $O(\sqrt{n})$ on planar graphs. We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width $O(1)$. Motivated by applications, we introduce a restricted version of the problem that we call the $\ell$-round General Propagation problem. We give a PTAS for the $\ell$-round General Propagation problem on planar graphs, for small values of $\ell$. Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes. In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph $G$ and a subset of terminal nodes $R\subseteq V(G)$. The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans $R$, that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only $3$ terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with $3$ terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of $\frac{1}{2}-\frac{1}{k}$, given a planar graph that is $k$ element-connected on the terminals; in fact, given such a graph the algorithm returns $k/2-1$ element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) $4$ for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio of $2-\epsilon$ in planar graphs.

This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees. In the Propagation problem we are given a graph $G$ and the goal is to find a minimum-sized set of nodes $S$ that covers all of the nodes, where a node $v$ is covered if (1) $v$ is in $S$, or (2) $v$ has a neighbor $u$ such that $u$ and all of its neighbors except $v$ are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use $n$ to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in weighted (general) graphs. The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem. We show that it is hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in general graphs. We design and analyze an approximation algorithm with a performance guarantee of $O(\sqrt{n})$ on planar graphs. We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width $O(1)$. Motivated by applications, we introduce a restricted version of the problem that we call the $\ell$-round General Propagation problem. We give a PTAS for the $\ell$-round General Propagation problem on planar graphs, for small values of $\ell$. Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes. In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph $G$ and a subset of terminal nodes $R\subseteq V(G)$. The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans $R$, that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only $3$ terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with $3$ terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of $\frac{1}{2}-\frac{1}{k}$, given a planar graph that is $k$ element-connected on the terminals; in fact, given such a graph the algorithm returns $k/2-1$ element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) $4$ for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio of $2-\epsilon$ in planar graphs.