In an alternative approach to \characterizing" the graph class of visibility graphs of simple polygons, we study the problem of nding a maximum clique in the visibility graph of a simple polygon with n vertices. We show that this problem is very hard, if the input polygons are allowed to contain holes: a gap-preserving reduction from the maximum clique problem on general graphs implies that no ...

In an alternative approach to “characterizing” the graph class of visibility graphs of simple polygons, we study the problem of finding a maximum clique in the visibility graph of a simple polygon with n vertices. We show that this problem is very hard, if the input polygons are allowed to contain holes: a gap-preserving reduction from the maximum clique problem on general graphs implies that n...

Parameterized complexity theory has enabled a refined classification of the difficulty of NPhard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH)...

A strong clique in a graph is intersecting all inclusion-maximal stable sets. Strong cliques play an important role the study of perfect graphs. We class diamond-free graphs, from both structural and algorithmic points view. show that following five NP -hard or co-NP problems remain when restricted to graphs: Is given strong? Does have clique? every vertex contained Given partition set into cli...

We provide an extensive study of the differential properties of the functions x 7→ x t −1 over F2n , for 1 < t < n. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x t + bx + (b + 1)x where b varies in F2n .We prove a strong relationship between the differential spectra of x 7→ x t −1 and x 7→ x s −1 for s = n− t+ ...

In 1965,Motzkin and Straus established a remarkable connection between the global maxima of the Lagrangian of a graph G over the standard simplex and the clique number of G. In this paper, we provide a generalization of the Motzkin–Straus theorem to k-uniform hypergraphs (k-graphs). Specifically, given a k-graph G, we exhibit a family of (parameterized) homogeneous polynomials whose local (glob...

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. We prove in this paper that the PATHWIDTH problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is polynomial.

For a graph G, we show a theorem that establishes a correspondence between the fine Hilbert series of the Stanley-Reisner ring of the clique complex for the complementary graph of G and the fine subgraph polynomial of G. We obtain from this theorem some corollaries regarding the independent set complex and the matching complex.

A graph polynomial p(G, X̄) can code numeric information about the underlying graph G in various ways: as its degree, as one of its specific coefficients or as evaluations at specific points X̄ = x̄0. In this paper we study the question how to prove that a given graph parameter, say ω(G), the size of the maximal clique of G, cannot be a fixed coefficient or the evaluation at any point of the Tutte...

Yannakakis’ Clique versus Independent Set problem (CL− IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size O(n), and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a p...