Abstract. 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.
Abstract. We consider the problem of finding, for two pairs (s1, t1) and (s2, t2) of vertices in an undirected graphs, an (s1, t1)-path P1 and an (s2, t2)-path P2 such that P1 and P2 share no edges and the length of each Pi satisfies Li, where Li ∈ {≤ ki, = ki, ≥ ki, ≤ ∞}. We regard k1 and k2 as parameters and investigate the parameterized complexity of the above problem when at least one of P1 and P2 has a length constraint (note that Li = " ≤ ∞" indicates that Pi has no length constraint). For the nine different cases of (L1, L2), we obtain FPT algorithms for seven of them. Our algorithms uses random partition backed by some structural results. On the other hand, we prove that the problem admits no polynomial kernel for all nine cases unless N P ⊆ coN P/poly.
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