On inclusionwise maximal and maximum cardinality $k$-clubs in graphs

Abstract

A $k$-club is a distance-based graph-theoretic generalization of a clique, originally introduced to model cohesive social subgroups in social network analysis. The $k$-clubs represent low diameter clusters in graphs and are appropriate for various graph-based data mining applications. Unlike cliques, the $k$-club model is nonhereditary, meaning every subset of a $k$-club is not necessarily a $k$-club. In this article, we settle an open problem establishing the intractability of testing inclusion-wise maximality of $k$-clubs. This result is in contrast to polynomial-time verifiability of maximal cliques, and is a direct consequence of its nonhereditary nature. We also identify a class of graphs for which this problem is polynomial-time solvable. We propose a distance coloring based upper-bounding scheme and a bounded enumeration based lower-bounding routine and employ them in a combinatorial branch-and-bound algorithm for finding maximum cardinality $k$-clubs. Computational results from using the proposed algorithms on 200-vertex graphs are also provided.