This paper introduces and studies the maximum k-plex problem, which arises in social network analysis, but can also be used in several other important application areas, including wireless networks, telecommunications, and graph-based data mining. We establish NP-completeness of the decision version of the problem on arbitrary graphs. An integer programming formulation is presented and basic polyhedral study of the problem is carried out. A branch-and-cut implementation is discussed and computational test results on the proposed benchmark instances and real-life scale-free graphs are also provided.

The maximum clique problem provides a classic framework for detecting cohesive subgraphs. However, this approach can fail to detect much of the cohesive structure in a graph.

Given a graph with non-negative edge weights, the MAX-CUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in different subsets is maximized. This classical NP-hard problem finds applications in VLSI design, statistical physics, and classification among other fields. This paper compares the performance of several greedy construction heuristics for MAX-CUT problem. In particular, a new "worst-out" approach is studied and the proposed edge contraction heuristic is shown to have an approximation ratio of at least 1/3. The results of experimental comparison of the worst-out approach, the well-known best-in algorithm, and modifications for both are also included.

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps.Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing.Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.Connected zero forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph. The connected zero forcing number of a graph is the cardinality of the smallest connected set of initially colored vertices which forces the entire graph to be colored (i.e., the smallest connected zero forcing set). Applications and various structural and computational aspects of connected zero forcing have been investigated in [15,16,17]; in particular, it can be used for modeling the spread of ideas or diseases originating from a single connected source in a network, or for power network monitoring accounting for the cost of supporting infrastructure. Other variants of zero forcing, such as positive semidefinite zero forcing [11,33,44,57], fractional zero forcing, signed zero forcing [41], and k-forcing [5,48] have also been studied. These are typically obtained by modifying the zero forcing color change rule, or adding certain restrictions to a zero forcing set. The number of timesteps in the zero forcing process after which a graph becomes colored is also a problem of interest (see, e.g., [14,21,28,43,57]). Connected variants of other graph problems -such as connected domination and connected power domination [25,31,36,40] -have been extensively studied as well.A closely related problem to zero forcing is power domination, where given a set S of initially colored vertices, the zero forcing color change rule is applied to N [S] instead of to S. Integer programming formulations for power domination and its variants have been explored in [1,18].The power domination problem is derived from the phase measurement unit (PMU) placement problem in electrical engineering, which has also been studied extensively; see, e.g., [47,49] and the bibliographies therein for various integer programming models and combinatorial algorithms for the PMU placement problem. Another closely related problem to zero forcing is the target set selection problem, where given a set S of initially colored vertices and a threshold function θ : V (G) → Z, all uncolored vertices v that have at least θ(v) colored neighbors become colored. Thus, the zero forcing problem constrains the infectors, but the target set selection problem constrai...