Co-2-plex vertex partitions

Abstract

This paper studies co-k-plex vertex partitions and more specifically co-2-plex vertex partitions. Co-$$k$$k-plexes and $$k$$k-plexes were first introduced in 1978 in the context of social network analysis. However, the study of co-k-plex vertex partitions or decomposing a graphs into degree bounded subgraphs can be at least dated back to the work of Lovasz (Studia Sci Math Hung 1:237---238, 1966). In this paper, we derive analogues for well-known results on the chromatic number, and present two algorithms for constructing co-2-plex vertex partitions. The first algorithm minimizes the number of partition classes while the second algorithm minimizes a weighted sum of the partition classes, where the weight of a partition class depends on the level of adjacency among its vertices.