In this paper, we present how 3D split and merge segmentation using topological and geometrical structuring with an Oriented Boundary Graph may be optimized by parallel algorithms. This structuring allows to implement efficiently split and merge operations, but since these treatments have often to be applied with large images, we have studied how to improve performances by parallelizing this pr...

We introduce a graph structure \(\Gamma^{\ast}_2(R)\) for commutative rings with unity. study some of the properties \(\Gamma^{\ast}_2(R)\). Also we parameters and find which is split.

MOTIVATION In recent years graph-theoretic descriptions have been applied to aid the analysis of a number of complex biological systems. However, such an approach has only just begun to be applied to examine protein structures and the network of interactions between residues with promising results. Here we examine whether a graph measure known as closeness is capable of predicting regions where...

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge c...

For fixed integers r, ` ≥ 0, a graph G is called an (r, `)-graph if the vertex set V (G) can be partitioned into r independent sets and ` cliques. This brings us to the following natural parameterized questions: Vertex (r, `)-Partization and Edge (r, `)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set...

For a graph G, a detachment operation at a vertex transforms the graph into a new graph by splitting the vertex into several vertices in such a way that the original graph can be obtained by contracting all the split vertices into a single vertex. A graph obtained from a given graph G by applying detachment operations at several vertices is called a detachment of graph G. We consider a detachme...

A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the graph. In this work we study the problem of finding a minimal connected tropical subgraph. We first show that this problem is NP-Hard for trees, interval graphs and split graphs, but polynomial when the number of colors is logarithmic in terms of the order of the graph (i.e. FPT). We then provide ...

We use the theory of oriented matroids to show that any linear embedding of K9, the complete graph on nine vertices, contains a non-split link with three components.

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of P, and Ept(P) is the EPT graph (i.e. the edge intersection graph) of the paths P of the tree T . These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least...