Dual Connectedness of Edge-Bicolored Graphs and Beyond.

Let G be an edge-bicolored graph where each edge is colored either red or blue. We study problems of obtaining an induced subgraph H from G that simultaneously satisfies given properties for H's red graph and blue graph. In particular, we consider Dually Connected Induced Subgraph problem - find from G a k-vertex induced subgraph whose red and blue graphs are both connected, and Dual Separator problem - delete at most k vertices to simultaneously disconnect red and blue graphs of G. We will discuss various algorithmic and complexity issues for Dually Connected Induced Subgraph and Dual Separator problems: NP-completeness, polynomial-time algorithms, W[1]-hardness, and FPT algorithms. As by-products, we deduce that it is NP-complete and W[1]- hard to find k-vertex (resp., (n - k)-vertex) strongly connected induced subgraphs from n-vertex digraphs. We will also give a complete characterization of the complexity of the problem of obtaining a k-vertex induced subgraph H from G that simultaneously satisfies given hereditary properties for H's red and blue graphs.