The Graph Cuts method of interactive segmentation has become very popular in recent years. This method performs at interactive speeds for smaller images/volumes, but an unacceptable amount of storage and computation time is required for the large images/volumes common in medical applications. The Banded Graph Cut (BGC) algorithm was proposed to drastically increase the computational speed of Gr...

We present a fully dynamic algorithm that maintains three different representations of an interval graph: a minimal interval model of the graph, the PQ-tree of its maximal cliques, and its modular decomposition. After each vertex or edge modification (insertion or deletion), the algorithm determines whether the new graph is an interval graph in O(n) time, and, in the positive, updates the three...

In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polyno...

The aim of edge modification problems is to change the edge set of a given graph as little as possible in order to satisfy a certain property. Edge modification problems in graphs have a lot of applications in different areas, and many polynomial-time algorithms and NP-completeness proofs for this kind of problems are known. In this work we prove new NP-completeness results for these problems i...

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline g...

Graph modification problems such as vertex deletion, edge deletion or edge contractions are a fundamental class of optimization problems. Recently, the parameterized complexity of the contractibility problem has been pursued for various specific classes of graphs. Usually, several graph modification questions of the deletion variety can be seen to be FPT if the graph class we want to delete int...

The Ord’s graph is a simple graphical method for displaying frequency distributions of data or theoretical distributions in the two-dimensional plane. Its coordinates are proportions of the first three moments, either empirical or theoretical ones. A modification of the Ord’s graph based on proportions of indices of qualitative variation is presented. Such a modification makes the graph applica...

In this paper, we solve the dynamic recognition problem for the class of P4-sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognizeif each such modification yields a P4-sparse graph, and if yes, to update a representationof the graph. Our approach relies on maintaining the modular decomposition tree of thegraph, which we use for solving the re...