Given an undirected graph G = (V,E) and a nonnegative integer k, the NPhard Cluster Editing problem asks whether G can be transformed into a disjoint union of cliques by modifying at most k edges. In this work, we study how “local degree bounds” influence the complexity of Cluster Editing and of the related Cluster Deletion problem which allows only edge deletions. We show that even for graphs ...

Consider the edge-deletion process in which the edges of some finite tree T are removed one after the other in the uniform random order. Roughly speaking, the cut-tree then describes the genealogy of connected components appearing in this edge-deletion process. Our main result shows that after a proper rescaling, the cut-tree of a critical Galton-Watson tree with finite variance and conditioned...

Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

The chromatic vertex (resp. edge) stability number vsχ(G) esχ(G)) of a graph G is the minimum vertices edges) whose deletion results in H with χ(H)=χ(G)−1. In main result it proved that if χ(G)∈{Δ(G),Δ(G)+1}, then vsχ(G)=ivsχ(G), where ivsχ(G) independent number. need not hold for graphs χ(G)≤Δ(G)+12. It χ(G)>Δ(G)2+1, vsχ(G)=esχ(G). A Nordhaus–Gaddum-type on also given.

Let be a simple and connected graph. A k-vertex separator [k-edge separator] is a subset of vertices [edges] whose deletion separates the vertex [edge] set of into two parts of equal cardinality, that are at distance greater than k in . Here we investigate the relation between the cardinality of these cutsets and the laplacian spectrum of . As a consequence of the study, we obtain the well-know...

Several authors have studied the graphs for which every edge is a chord of a cycle; among 2-connected graphs, one characterization is that the deletion of one vertex never creates a cut-edge. Two new results: among 3-connected graphs with minimum degree at least 4, every two adjacent edges are chords of a common cycle if and only if deleting two vertices never creates two adjacent cut-edges; am...

This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of curves. We give several formulas for the tau constant, and show how it changes under graph operations including deletion of an edge, contraction of an edge, and union of graphs along one or two points. We show how the tau constant changes when edges o...

In this paper, we discuss the problem of reoptimization of Steiner tree. We are given an instance of Graph and also an optimal Steiner tree of it. If some changes occurs later on in the given graph. New optimal Steiner tree is to be determine, this process is known as re optimization. We consider two cases of change: one is addition of a new edge and second is, Deletion of an existing edge. For...

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...

 Background: Spinal muscular atrophy (SMA) is the second most common lethal autosomal recessive disease. It is a neuromuscular disorder caused by degenerative of lower motor neurons and occasionally bulbar neurons leading to progressive limb paralysis and muscular atrophy. The SMN1 gene is recognized as a SMA causing gene while NAIP has been characterized as a modifying factor for the clinical ...