We propose a polynomial-time algorithm which takes as input a finite set of points of R3 and compute, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii. Almost three decades ago, an elegant...

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A cliqueindependent set is a subset of pairwise disjoint cliques of G. Denote by τC(G) and αC(G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τC(H) = αC(H), for every induced subgraph H of G. In this paper, w...

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lovász and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical ri...

The spectral radius (or the signless Laplacian radius) of a general hypergraph is maximum modulus eigenvalues its adjacency Laplacian) tensor. In this paper, we firstly obtain lower bound hypergraphs in terms clique number. Moreover, present relation between homogeneous polynomial and number hypergraphs. As an application, finally upper

It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1− , for any constant > 0, unless NP = ZPP. In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that unless NP ⊆ ZPTIME(2 logn) 3/2)), neither Max Clique nor Min Chro...

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is deenable in Monadic Second Order Logic. We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition , which establishes the bound on the clique{width, can be computed in polynomia...

A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, solving on special classes, it is helpful to know if the class under consideration has bounded width. In this paper we consider maximum-induced matching (mim-width), a particularly general parameter that algorithmic applications whenever decomposition “quickly computable” for cons...

In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K ∩S = ∅, there exists a cut (W,W ′) in C such that K ⊆W and S ⊆W ′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [20] asked in 1...

The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the “kernelization” of Leaf Out Branching(k) into...