The reducibility of optimal 1-planar graphs with respect to the lexicographic product

A graph is called 1-planar if it can be drawn on the plane (or on the sphere) such that each edge is crossed at most once. A 1-planar graph $G$ is called optimal if it satisfies $|E(G)| = 4|V(G)|-8$. If $G$ and $H$ are graphs, then the lexicographic product $G\circ H$ has vertex set the Cartesian product $V(G)\times V(H)$ and edge set $\{(g_1,h_1) (g_2,h_2): g_1 g_2 \in E(G),\,\, \text{or}\,\, g_1=g_2 \,\, \text{and}\,\, h_1 h_2 \in E(H)\}$. A graph is called reducible if it can be expressed as the lexicographic product of two smaller non-trivial graphs. In this paper, we prove that an optimal 1-planar graph $G$ is reducible if and only if $G$ is isomorphic to the complete multipartite graph $K_{2,2,2,2}$. As a corollary, we prove that every reducible 1-planar graph with $n$ vertices has at most $4n-9$ edges for $n=6$ or $n\ge 9$. We also prove that this bound is tight for infinitely many values of $n$. Additionally, we give two necessary conditions for a graph $G\circ 2K_1$ to be 1-planar.Comment: 23 pages, 14 fugure

Computational Complexity And Algorithms For Dirty Data Evaluation And Repairing

In this dissertation, we study the dirty data evaluation and repairing problem in relational database. Dirty data is usually inconsistent, inaccurate, incomplete and stale. Existing methods and theories of consistency describe using integrity constraints, such as data dependencies. However, integrity constraints are good at detection but not at evaluating the degree of data inconsistency and cannot guide the data repairing. This dissertation first studies the computational complexity of and algorithms for the database inconsistency evaluation. We define and use the minimum tuple deletion to evaluate the database inconsistency. For such minimum tuple deletion problem, we study the relationship between the size of rule set and its computational complexity. We show that the minimum tuple deletion problem is NP-hard to approximate the minimum tuple deletion within 17/16 if given three functional dependencies and four attributes involved. A near optimal approximated algorithm for computing the minimum tuple deletion is proposed with a ratio of 2 − 1/2r , where r is the number of given functional dependencies. To guide the data repairing, this dissertation also investigates the data repairing method by using query feedbacks, formally studies two decision problems, functional dependency restricted deletion and insertion propagation problem, corresponding to the feedbacks of deletion and insertion. A comprehensive analysis on both combined and data complexity of the cases is provided by considering different relational operators and feedback types. We have identified the intractable and tractable cases to picture the complexity hierarchy of these problems, and provided the efficient algorithm on these tractable cases. Two improvements are proposed, one focuses on figuring out the minimum vertex cover in conflict graph to improve the upper bound of tuple deletion problem, and the other one is a better dichotomy for deletion and insertion propagation problems at the absence of functional dependencies from the point of respectively considering data, combined and parameterized complexities

On the Basis Number of the Strong Product of Theta Graphs with Cycles

In graph theory, there are many numbers that give rise to a better understanding and interpretation of the geometric properties of a given graph such as the crossing number, the thickness, the genus, the basis number, etc.

Graphs of Polytopes

The graph of a polytope is the graph whose vertex set is the set of vertices of the polytope, and whose edge set is the set of edges of the polytope. Several problems concerning graphs of polytopes are discussed. The primary result is a set of bounds (Theorem 39) on the maximal size of an anticlique (sometimes called a coclique, stable set, or independent set) of the graph of a polytope based on its dimension and number of vertices. Two results concerning properties preserved by certain operations on polytopes are presented. The first is that the Gale diagram of a join of polytopes is the direct sum of the Gale diagrams of the polytopes and dually, that the Gale diagram of a direct sum of polytopes is the join of their Gale diagrams (Theorem 23). The second is that if two polytopes satisfy a weakened form of Gale's evenness condition, then so does their product (Theorem 32). It is shown, by other means, that, with only two exceptions, the complete bipartite graphs are never graphs of polytopes (Theorem 47). The techniques developed throughout are then used to show that the complete 3-partite graph K_{1,n,m} is the graph of a polytope if and only if K_{n,m} is the graph of a polytope (Theorem 49). It is then shown that K_{2,2,3} and K_{2,2,4} are never graphs of polytopes. A conjecture is then stated as to precisely when a complete multipartite graph is the graph of a polytope. Finally, a section is devoted to results concerning the dimensions for which the graph of a crosspolytope is the graph of a polytope