The Correlation Clustering problem, also known as the Cluster Editing problem, seeks to edit a given graph by adding and deleting edges to obtain a collection of vertex-disjoint cliques, such that the editing cost is minimized. The Edge Clique Partitioning problem seeks to partition the edges of a given graph into edge-disjoint cliques, such that the number of cliques is minimized. Both problem...

A Fixed-Parameter Tractable (FPT) ρ-approximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPT-algorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Cliq...

Structural and behavioral parameters of many real networks such as social networks are unpredictable, uncertain and time varying parameters and for these reasons deterministic graphs for modeling such networks are too restrictive to solve most of the real network problems. It seems that stochastic graphs, in which weights associated to the vertices are random variable, may be a better graph mod...

In Fixed Parameter Tractability (FPT) theory, we are given a problem with a special parameter k. In this paper we are only interested in k equal the size of the optimum. A FPT algorithm for a problem is an exact algorithm that runs in time h(k) · nO(1) for a function h that may be arbitrarily large. In FPT approximation we seek a g(k) ratio that runs in time h(k) · nO(1), so that h,g are two in...

A clique of an undirected graph G (V, E) with a vertex set V and an edge set E is a subset of V such that all pairs of vertices are connected by an edge in E. The maximum clique problem (MCP) is to find a clique of maximum size of the graph G. Figure 1(b) shows a maximum clique of the graph Figure 1(a) with 10 vertices and 21 edges. It is one of the first problems which have been proven to be N...

The Correlation Clustering problem, also known as the Cluster Editing problem, seeks to edit a given graph by adding and deleting edges to obtain a collection of vertex-disjoint cliques, such that the editing cost is minimized. The Edge Clique Partitioning problem seeks to partition the edges of a given graph into edge-disjoint cliques, such that the number of cliques is minimized. Both problem...

we give a new recursive method to compute the number of cliques and cycles of a graph. this method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. in particular, let $g$ be a graph and let $overline {g}$ be its complement, then given the chromatic polynomial of...

This paper studies the maximum-clique independence problem and some variations of clique transversal such as {k}-clique, maximum-clique, minus clique, signed k-fold problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal perfect total split line dually graphs. We give equations to compute numbers show that {k}-clique is polynomial-time solvable graphs whose equal thei...