Let L3 be the class of edge intersection graphs of linear 3-uniform hypergraphs. The problem of recognizing G ∈ L3 is NP-complete. Denote by δALG the minimal integer such that the problem ”G ∈ L3 ” is polynomially solvable in the class of graphs G with the minimal vertex degree δ(G) ≥ δALG and by δFIS the minimal integer such that L3 can be characterized by a finite list of forbidden induced su...

The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a pmedian with respect to the new edge lengths. The problem is shown to be strongly NP-hard on general graphs and weakly NP-hard on series-parallel graphs. Therefore, the special case on...

For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...

In 2-neighborhood bootstrap percolation on a graphG, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least two already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to e...

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequen...

A graph is H-free if it does not contain an induced subgraph isomorphic to H . We denote by Pk and Ck the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on kcoloring Pt-free graphs, and almost complete the classificatio...

A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertex-colour a given graph, is known for a large number of graph classes. Notably it is NP-complete in general, but polynomial time solvable for perfect graphs. A triple of vert...

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Odd Cycle Transversal, by restricting input to H-free graphs, that is, graphs do not contain some fixed H as an induced subgraph. By combining known new results, we determine both problems on for every except when H=sP1+P4 s≥1. As part our approach, introduce Cover problem p...

The (asymptotic) dimension growth functions of groups were introduced by Gromov in 1999. In this paper, we show connections between dimension growth and expansion properties of graphs, Ramsey theory and the Kolmogorov-Ostrand dimension of groups and prove that all solvable subgroups of the R.Thompson group F have polynomial dimension growth. We introduce controlled dimension growth function and...