Graph Motif Problems Parameterized by Dual

Let G = (V,E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G, a multisetM of colors built from C, and asks whether there is a subset S ⊆ V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex v of V may choose its color from a list L(v) of colors. We study the three problems GM, CGM, and LGM, parameterized by ` := |V | − |M |. In particular, for general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no (2− ) ·|V |O(1)-time algorithm, which implies that a previous algorithm, running in O(2 · |E|) time is optimal [2]. We also prove that LGM is W[1]-hard even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in O(4 · |V |) time but admits no polynomial-size problem kernel, while CGM can be solved in O( √ 2 + |V |) time and admits a polynomial-size problem kernel. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory