Cluster Deletion and Cluster Editing ask to transform a graph by at most k edge deletions or edge edits, respectively, into a cluster graph, i.e., disjoint union of cliques. Equivalently, a cluster graph has no conflict triples, i.e., two incident edges without a transitive edge. We solve the two problems in time O∗(1.415k) and O∗(1.76k), respectively. These results round off our earlier work b...

This paper presents a three-module strategy for edge detection. The first and the last module involve well-known methods: the first module is a parallel process computing local edge strength and direction while the last module is a sequential process following edges. The originality of the overall method resides in the intermediate module, seen as a generalization of the nonmaximum deletion alg...

Given a graph G and an integer k, the Π Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property Π . Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tr...

We present efficient fixed-parameter algorithms for the NP-complete edge modification problems Cluster Editing and Cluster Deletion. Here, the goal is to make the fewest changes to the edge set of an input graph such that the new graph is a vertex-disjoint union of cliques. Allowing up to k edge additions and deletions (Cluster Editing), we solve this problem in O(2.27 + |V |) time; allowing on...

K.Dohmen, A.Pönitz and P.Tittman (2003), introduced a bivariate generalization of the chromatic polynomial P (G, x, y) which subsumes also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We first show that P (G, x, y) has a recursive definition with respect to three kinds of edge elimi...

Editing a graph into disjoint union of clusters is standard optimization task in graph-based data clustering. Here, complementing classic work where the shall be cliques, we focus on that 2-clubs, is, subgraphs diameter two. This naturally leads to two NP-hard problems 2-Club Cluster (the allowed editing operations are edge insertion and deletion) Vertex Deletion vertex deletions). Answering an...

For a family of graphs F , a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F . F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G = (V,∪i=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colo...

In edge deletion problems, we are given a graph G and a graph property π and the task is to find a subset of edges the deletion of which results in a subgraph of G satisfying the property π. Typically the objective is to minimize the total number of deleted edges, while in less common fair versions the objective is to minimize the maximum number of edges removed from a single vertex. Since many...

In this paper, some new results are introduced for the bi-domination in graphs. Some properties of number and bounds according to maximum, minimum degrees, order, size have been determined. The effects removing a vertex or adding an edge discussed on graph. This study is important know affected graphs by deletion addition components.

We analyse the relations between several graph transformations that were introduced to be used in procedures determining the stability number of a graph. We show that all these transformations can be decomposed into a sequence of edge deletions and twin deletions. We also show how some of these transformations are related to the notion of even pair introduced to color some classes of perfect gr...