Fast edge searching and fast searching on graphs

AbstractGiven a graph G=(V,E) in which a fugitive hides on vertices or along edges, graph searching problems are usually to find the minimum number of searchers required to capture the fugitive. In this paper, we consider the problem of finding the minimum number of steps to capture the fugitive. We introduce the fast edge searching problem in the edge search model, which is the problem of finding the minimum number of steps (called the fast edge-search time) to capture the fugitive. We establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model. While the family of graphs whose fast search number is at most k is not minor-closed for any positive integer k≥2, we show that the family of graphs whose fast edge-search time is at most k is minor-closed. We establish relations between the fast (fast edge) searching and the node searching. These relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge-search numbers or fast search numbers. Using these relations, we prove that the problem of deciding, given a graph G and an integer k, whether the fast (edge-)search number of G is less than or equal to k is NP-complete; and it remains NP-complete for Eulerian graphs. We also prove that the problem of determining whether the fast (edge-)search number of G is half of the number of odd vertices in G is NP-complete; and it remains NP-complete for planar graphs with maximum degree 4. We present a linear time approximation algorithm for the fast edge-search time that always delivers solutions of at most (1+|V|−1|E|+1) times the optimal value. This algorithm also gives us a tight upper bound on the fast search number of graphs. We also show a lower bound on the fast search number using the minimum degree and the number of odd vertices

Studies of several tetrahedralization problems

The main purpose of decomposing an object into simpler components is to simplify a problem involving the complex object into a number of subproblems having simpler components. In particular, a tetrahedralization is a partition of the input domain in R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations have applications in the finite element method, mesh generation, computer graphics, and robotics. This thesis investigates four problems in tetrahedralizations and triangulations. The first problem is on the computational complexity of tetrahedralization detections. We present an O(nm log n) algorithm to determine whether a set of line segments .C is the edge set of a tetrahedralization, where m is the number of segments and n is the number of endpoints in .C. We show that it is NP-complete to decide whether .C contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations. After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized in O(n²) time. The third problem is on the tetrahedralization of two nested convex polyhedra. We give a method to tetrahedralize the region between two nested convex polyhedra into a linear number of tetrahedra without introducing Steiner points. This result answers an open problem raised by Bern [16]. The fourth problem is on the lower bound for β-skeletons belonging to minimum weight triangulations. We prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value, the β-skeleton of a point set may not always be a subgraph of the minimum weight triangulation of this point set. This result settles Keil's conjecture [62]

Sweeping graphs with large clique number

AbstractSearching a network for intruders is an interesting and often difficult problem. Sweeping (or edge searching) is one such search model, in which intruders may exist anywhere along an edge. It was conjectured that graphs exist for which the connected sweep number is strictly less than the monotonic connected sweep number. We prove that this is true, and the difference can be arbitrarily large. We also show that the clique number is a lower bound on the sweep number

Genomic Scaffold Filling Revisited

The genomic scaffold filling problem has attracted a lot of attention recently. The problem is on filling an incomplete sequence (scaffold) I into I\u27, with respect to a complete reference genome G, such that the number of adjacencies between G and I\u27 is maximized. The problem is NP-complete and APX-hard, and admits a 1.2-approximation. However, the sequence input I is not quite practical and does not fit most of the real datasets (where a scaffold is more often given as a list of contigs). In this paper, we revisit the genomic scaffold filling problem by considering this important case when, (1) a scaffold S is given, the missing genes X = c(G) - c(S) can only be inserted in between the contigs, and the objective is to maximize the number of adjacencies between G and the filled S\u27 and (2) a scaffold S is given, a subset of the missing genes X\u27 subset X = c(G) - c(S) can only be inserted in between the contigs, and the objective is still to maximize the number of adjacencies between G and the filled S\u27\u27. For problem (1), we present a simple NP-completeness proof, we then present a factor-2 greedy approximation algorithm, and finally we show that the problem is FPT when each gene appears at most d times in G. For problem (2), we prove that the problem is W[1]-hard and then we present a factor-2 FPT-approximation for the case when each gene appears at most d times in G

Narrowing the Complexity Gap for Colouring (C_s,P_t)-Free Graphs

Let k be a positive integer. The k-Colouring problem is to decide whether a graph has a k-colouring. The k-Precolouring Extension problem is to decide whether a colouring of a subset of a graph’s vertex set can be extended to a k-colouring of the whole graph. A k-list assignment of a graph is an allocation of a list — a subset of {1,…,k} — to each vertex, and the List k -Colouring problem asks whether the graph has a k-colouring in which each vertex is coloured with a colour from its list. We prove a number of new complexity results for these three decision problems when restricted to graphs that do not contain a cycle on s vertices or a path on t vertices as induced subgraphs (for fixed positive integers s and t)