Covering with Clubs: Complexity and Approximability

Lecture Notes in Computer Science book series (LNCS, volume 10979)Finding cohesive subgraphs in a network is a well-known problem in graph theory. Several alternative formulations of cohesive subgraph have been proposed, a notable example being s-club, which is a subgraph where each vertex is at distance at most s to the others. Here we consider the problem of covering a given graph with the minimum number of s-clubs. We study the computational and approximation complexity of this problem, when s is equal to 2 or 3. First, we show that deciding if there exists a cover of a graph with three 2-clubs is NP-complete, and that deciding if there exists a cover of a graph with two 3-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of 2-clubs and 3-clubs. We show that, given a graph G=(V,E) to be covered, covering G with the minimum number of 2-clubs is not approximable within factor O(|V|1/2−ε), for any ε>0, and covering G with the minimum number of 3-clubs is not approximable within factor O(|V|1−ε), for any ε>0. On the positive side, we give an approximation algorithm of factor 2|V|1/2log3/2|V| for covering a graph with the minimum number of 2-clubs

Domination Related Parameters and Applications

In this chapter, we explore two graph theoretical concepts introduced by Stephen Hedetniemi as models for real-life applications. The first, Roman domination, is based on a historical account of a defense strategy used by the Roman Empire; and the second, alliances in graphs, models an agreement between two or more parties to work together for the common good

Paired Domination in Graphs

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The minimum cardinality of a paired dominating set of G is the paired domination number of G. This chapter presents a survey of the major results on paired domination with an emphasis on bounds for the paired domination number