Dieter Kratsch

Treewidth and minimum fill-in on d-trapezoid graphs

We show that the minimum ll-in and the minimum interval graph completion of a d-trapezoid graph can be computed in time O(n d ). We also show that the treewidth and the pathwidth of a d-trapezoid graph can be computed in time O(ntw(G) d 1 ). In both cases, d is supposed to be a xed positive integer and it is required that a suitable intersection model of the given d-trapezoid graph is part of the input. As a consequence, each of the four graph parameters can be computed in time O(n 2 ) for trapezoid graphs and thus for permutation graphs even if no intersection model is part of the input.

Minimal dominating sets in graph classes: Combinatorial bounds and enumeration

Abstract The number of minimal dominating sets that a graph on n vertices can have is known to be at most 1.715 9 n . This upper bound might not be tight, since no examples of graphs with 1.570 5 n or more minimal dominating sets are known. For several classes of graphs, we substantially improve the upper bound on the number of minimal dominating sets. At the same time, we give algorithms for enumerating all minimal dominating sets, where the running time of each algorithm is within a polynomial factor of the proved upper bound for the graph class in question. In several cases, we provide examples of graphs containing the maximum possible number of minimal dominating sets for graphs in that class, thereby showing the corresponding upper bounds to be tight.

Finding Shortest Paths Between Graph Colourings

The

Subset feedback vertex sets in chordal graphs

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Parameterized Algorithms for Finding Square Roots

k-colouring reconfiguration problem asks whether, for a given graph

Enumerating minimal dominating sets in chordal bipartite graphs

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