Fixed parameter tractability of crossing minimization of almost-trees

We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.Comment: Graph Drawing 201

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

Efficiently Realizing Interval Sequences

We consider the problem of realizable interval-sequences. An interval sequence comprises of $n$ integer intervals $[a_i,b_i]$ such that $0\leq a_i \leq b_i \leq n-1$, and is said to be graphic/realizable if there exists a graph with degree sequence, say, $D=(d_1,\ldots,d_n)$ satisfying the condition $a_i \leq d_i \leq b_i$, for each $i \in [1,n]$. There is a characterisation (also implying an $O(n)$ verifying algorithm) known for realizability of interval-sequences, which is a generalization of the Erdos-Gallai characterisation for graphic sequences. However, given any realizable interval-sequence, there is no known algorithm for computing a corresponding graphic certificate in $o(n^2)$ time. In this paper, we provide an $O(n \log n)$ time algorithm for computing a graphic sequence for any realizable interval sequence. In addition, when the interval sequence is non-realizable, we show how to find a graphic sequence having minimum deviation with respect to the given interval sequence, in the same time. Finally, we consider variants of the problem such as computing the most regular graphic sequence, and computing a minimum extension of a length $p$ non-graphic sequence to a graphic one.Comment: 19 pages, 1 figur

The Graph Motif problem parameterized by the structure of the input graph

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. Though, in the many applications of Graph Motif, the input graph originates from real-life 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.Comment: 24 pages, accepted in DAM, conference version in IPEC 201