Recently, D. Corneil found a simple 3-sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3-sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m + n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n), the certificates of non-representability (the forbidden subgraphs) can be authenticated in time O(n). 1

We present a simple exact algorithm for the INDEPENDENT SET problem with a runtime bounded by O(1.2132 npoly(n)). This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computer-aided case analysis. The new branching rule uses the concept of satellites and has previously only been used in an algorithm for sparse graphs. The computer-aided case analysis allows us to capture the behavior of our algorithm in more detail than in a traditional analysis. The main purpose of this paper is to demonstrate how a very simple algorithm can outperform more complicated ones if the right analysis of its running time is performed.

We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the NP-hard problem to delete a minimum number of rows or columns from a 0/1-matrix such that the remaining submatrix has the C1P.

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We survey the consecutive-ones property of binary matrices. Herein, a binary matrix has the consecutive-ones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.

When I was asked to contribute to a volume dedicated to Thomas Ottmann’s sixtieth birthday, I

Abstract. We provide a certifying algorithm for the problem of deciding whether a P5-free graph is 3-colorable by showing there are exactly six finite graphs that are P5-free and not 3-colorable and minimal with respect to this property. 1

We present an algorithm that supports operations for modifying a split graph by adding edges or vertices and deleting edges, such that after each modification the graph is repaired to become a split graph in a minimal way. In particular, if the graph is not split after the modification, the algorithm computes a minimal, or if desired even a minimum, split completion or deletion of the modified graph. The motivation for such operations is similar to the motivation for fully dynamic algorithms for particular graph classes. In our case we allow all modifications to the graph and repair, rather than allowing only modifications that keep the graph split. Fully dynamic algorithms of the latter kind are known for split graphs [24]. Our results can

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