A “double star” is a tree with two internal vertices. It known that the Gyárfás–Sumner conjecture holds for double stars, is, every star H $H$ , there function f ${f}_{H}$ such if G $G$ does not contain as an induced subgraph then ? ( ) ? ? $\chi (G)\le {f}_{H}(\omega (G))$ (where ,\omega $ are chromatic number and clique of ). Here we prove can be chosen to polynomial.

• Assume that the input is random, and find an algorithm that will perform well in the average case. For example, the maximum clique problem, which is NP -hard, can actually be solved efficiently assuming a random input because the maximum clique in a randomly chosen graph is small. This assumption is often used in practice, but the problem is that not everyone will agree on whether the input d...

Biró et al. (Discrete. Math 100(1–3):267–279, 1992) introduced the concept of H-graphs, intersection graphs connected subgraphs a subdivision graph H. They are related to and generalize many important classes geometric graphs, e.g., interval circular-arc split chordal graphs. Our paper starts new line research in area by studying several classical computational problems on H-graphs: recognition...

We give a reduction from clique to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Here we present polynomial-time combinatorial algorithms for the optimal weighted coloring and weighted clique problems in bull-free perfect graphs. The algorithms are based on a structural analysis and decomposition of bull-free perfect graphs.

A graph is biclique-Helly when its family of (maximal) bicliques is a Helly family. We describe characterizations for biclique-Helly graphs, leading to polynomial time recognition algorithms. In addition, we relate biclique-Helly graphs to the classes of clique-Helly, disk-Helly and neighborhood-Helly graphs.

In a signed graph G, a negative clique is a complete subgraph having negative edges only. In this article, we give characteristic polynomial expressions, and eigenvalues of some signed graphs having negative cliques. This includes signed cycle graph, signed path graph, a complete graph with disjoint negative cliques, and star block graph with negative cliques.

It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.

Neighbor-scattering number is a useful measure for graph vulnerability. For some special kinds of graphs, explicit formulas are given for this number. However, for general graphs it is shown that to compute this number is NP-complete. In this paper, we prove that for interval graphs this number can be computed in polynomial time. Keyworks: neighbor-scattering number, interval graph, consecutive...

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+ ...