Partitioning P4-tidy graphs into a stable set and a forest

Given a graph G, a near-bipartition of G is a partition of V(G) into S and F, where S is a stable set, and F induces a forest. Given a graph G and a vertex set P inducing a P4 in G, a vertex v is said partner of P if v∈V(G)∖P and G[P∪{v}] has at least two P4’s. A graph G is P4-tidy if any P4 in G has at most one partner. In this paper, we study the property of having a near-bipartition on P4-tidy graphs. We also consider a variant of that property that requires F being a tree. While the problem of determining whether an input graph has a near-bipartition is NP-complete even for graphs with maximum degree four, perfect graphs, graphs with diameter three, line graphs, and planar graphs, we show that the class of near-bipartite P4-tidy graphs admit finite forbidden induced subgraph characterization. In addition, we also present a characterization for P4-tidy graphs admitting a partition of its vertex set into a stable set and a tree.