Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more e...

Let G be a directed graph and G be its r-th power. We study different issues dealing with the number of arcs, or size, of G and G: given the order and diameter of a strongly connected digraph, what is its maximum size, and which are the graphs achieving this bound? What is the minimum size of the r-th power of a strongly connected digraph, and which are the graphs achieving this bound? Given al...

Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined k-leaf powers as the class of graphs G = (V,E) which has a k-leaf root T , i.e., T is a tree such that the vertices of G are exactly the leaves of T and two vertices in V are adjacent in G if and only if their distance in T is at most k. It is known that leaf powers are chordal graphs. Brandstädt and Le pr...

Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. In typical applications like social networks or biological networks, however, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We pr...

How will a virus propagate in a real network? Does an epidemic threshold exist for a finite powerlaw graph, or any finite graph? How long does it take to disinfect a network given particular values of infection rate and virus death rate? We answer the first question by providing equations that accurately model virus propagation in any network including real and synthesized network graphs. We pr...

We adapt the compact routing scheme by Thorup and Zwick to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the power-law graph model for a compact routing scheme. In particular, we pr...

Spectral clustering, while perhaps the most efficient heuristics for graph partitioning, has recently gathered bad reputation for failure over large-scale power law graphs. In this chapter we identify the abundance of small-size communities connected by long tentacles as the major obstacle for spectral clustering. These subgraphs hide the higher level structure and result in a highly degenerate...

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519–529]). A set S of v...

We prove new explicit inapproximability results for the Vertex Cover Problem on the Power Law Graphs and some functional generalizations of that class of graphs. Our results depend on special bounded degree amplifier constructions for those classes of graphs and could be also of independent interest.