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The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring , the problem of determining whether a given graph has Grundy number at least k . We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Coloring (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that Grundy Coloring can be solved in time O ∗ ( 2 . 44 3 n ) and Weak Grundy Coloring in time O ∗ ( 2 . 71 6 n ) on graphs of order n . While Grundy Coloring and Weak Grundy Coloring are known to be solvable in time O ∗ ( 2 O ( w k ) ) for graphs of treewidth w (where k is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O ∗ ( 2 o ( w log w ) ) . We also describe an O ∗ ( 2 2 O ( k ) ) algorithm for Weak Grundy Coloring , which is therefore FPT for the parameter k . Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for Grundy Coloring ). Although we do not know whether Grundy Coloring is in FPT , we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for Grundy Coloring and Weak Grundy Coloring on apex-minor graphs. In stark contrast with the two other problems, we show that Connected Grundy Coloring is NP -complete already for k = 7 colors. [ABSTRACT FROM AUTHOR]

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. In the many utilizations of Graph Motif , however, the input graph originates from real-life applications and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1] -hard (while in W[P] ) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way. [ABSTRACT FROM AUTHOR]

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G 1 and G 2 , one looks for a graph with the maximum number of vertices being both an induced subgraph of G 1 and G 2 . MCIS is among the most studied classical NP -hard problems. It remains NP -hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of Clique , it is W [ 1 ] -hard parameterized by the size of the solution. Being NP -hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by k : = vc ( G 1 ) + vc ( G 2 ) the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time 2 O ( k log ⁡ k ) . We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2 o ( k log ⁡ k ) . This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by k , unless NP ⊆ coNP / poly . Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study three connection games among the most widely played: havannah , twixt , and slither . We show that determining the outcome of an arbitrary input position is PSPACE -complete in all three cases. Our reductions are based on the popular graph problem generalized geography and on hex itself. We also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is W[1] -hard, short hex is FPT . Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games. [ABSTRACT FROM AUTHOR]