On three polynomial kernels of sequences for arbitrarily partitionable graphs

A graph G is arbitrarily partitionable if every sequence (n1, n2, ..., np) of positive integers summing up to |V (G)| is realizable in G, i.e. there exists a partition (V1, V2, ..., Vp) of V (G) such that Vi induces a connected subgraph of G on ni vertices for every i ∈ {1, 2, ..., p}. Given a family F(n) of graphs with order n ≥ 1, a kernel of sequences for F(n) is a set KF (n) of sequences summing up to n such that every member G of F(n) is arbitrarily partitionable if and only if every sequence of KF (n) is realizable in G. We herein provide kernels with polynomial size for three classes of graphs, namely complete multipartite graphs, graphs with about a half universal vertices, and graphs made up of several arbitrarily partitionable components. Our kernel for complete multipartite graphs yields a polynomial-time algorithm to decide whether such a graph is arbitrarily partitionable.