The dominating graph of a H has as its vertices all sets H, with an edge between two if one can be obtained from the other by addition or deletion single vertex H. In this paper we prove that any tree Hamilton path. We also show how result about binary strings leads to proof cycle on n path and only is not multiple 4.

In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the C...

We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject to edge insertions and deletions and that allows queries that test whether the graph is currently planar, or whether a potentia...

In the last lecture we talked about how to support dynamic graphs with respect to operations which included insertion and deletion of edges and looking for existence of path between two vertices. In this lecture we will talk about supporting all pair shortest paths along with addition, deletion and updation of edges in O(n2 log n) amortized time on the graphs with non negative real-valued edge ...

Background & Aims: The most significant cause of infertility in men is the genetic deletion in the azoospermia factor (AZF) region that is caused by the process of intra- and inter-chromosomal homologous recombination in amplicons. Homologous recombination could also result in partial deletions in AZF region. The aim of this research was to determine the association between the partial AZFc del...

We prove that local complementation and vertex deletion, operations from which vertexminors are defined, can simulate edge contractions. As an application, we prove that the rank-width of a graph is linearly bounded in term of its tree-width.

This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We...

We show that O(n) exchanging flips suffice to transform any edge-labelled pointed pseudo-triangulation into any other with the same set of labels. By using insertion, deletion and exchanging flips, we can transform any edge-labelled pseudo-triangulation into any other with O(n log c+h log h) flips, where c is the number of convex layers and h is the number of points on the convex hull.

We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of K updates in K · polylog(n) time, where n is the number of vertices in the gra...

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for γP,k(G − e), γP,k(G/e) and for γP,k(G − v) in terms of γP,k(G), and give examples for which these bounds are tight. We characterize all graphs for which γP,k(G− e) = γP,k(G) + 1 for any edge e...