### Graph Motif Problems Parameterized by Dual

Guillaume Fertin, Christian Komusiewicz
2020 Journal of Graph Algorithms and Applications
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 multiset M 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
more » ... ach 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 . We also prove that LGM is W-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.