A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.Comment: 24 page

Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions

AbstractWe study graph properties that admit an increasing, or equivalently decreasing, sequence of graphs on the same vertex set such that for any two consecutive graphs in the sequence their difference is a single edge. This is useful for characterizing and computing minimal completions and deletions of arbitrary graphs into having these properties. We prove that threshold graphs and chain graphs admit such sequences. Based on this characterization and other structural properties, we present linear-time algorithms both for computing minimal completions and deletions into threshold, chain, and bipartite graphs, and for extracting a minimal completion or deletion from a given completion or deletion. Minimum completions and deletions into these classes are NP-hard to compute

07211 Abstracts Collection -- Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes

From May 20 to May 25, 2007, the Dagstuhl Seminar 07211 ``Exact, Approximative, Robust and Certifying Algorithms on Particular Graph Classes\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Chordal Graphs are Fully Orientable

Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let m and M denote the minimum and the maximum of the number of dependent arcs over all acyclic orientations of G. We call G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable.Comment: 11 pages, 1 figure, accepted by Ars Combinatoria (March 26, 2010

Structured Decompositions: Structural and Algorithmic Compositionality

We introduce structured decompositions: category-theoretic generalizations of many combinatorial invariants -- including tree-width, layered tree-width, co-tree-width and graph decomposition width -- which have played a central role in the study of structural and algorithmic compositionality in both graph theory and parameterized complexity. Structured decompositions allow us to generalize combinatorial invariants to new settings (for example decompositions of matroids) in which they describe algorithmically useful structural compositionality. As an application of our theory we prove an algorithmic meta theorem for the Sub_P-composition problem which, when instantiated in the category of graphs, yields compositional algorithms for NP-hard problems such as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs