1,256 research outputs found
Decomposition, approximation, and coloring of odd-minor-free graphs
We prove two structural decomposition theorems about graphs excluding
a fixed odd minor H, and show how these theorems can
be used to obtain approximation algorithms for several algorithmic
problems in such graphs. Our decomposition results provide new
structural insights into odd-H-minor-free graphs, on the one hand
generalizing the central structural result from Graph Minor Theory,
and on the other hand providing an algorithmic decomposition
into two bounded-treewidth graphs, generalizing a similar result for
minors. As one example of how these structural results conquer difficult
problems, we obtain a polynomial-time 2-approximation for
vertex coloring in odd-H-minor-free graphs, improving on the previous
O(jV (H)j)-approximation for such graphs and generalizing
the previous 2-approximation for H-minor-free graphs. The class
of odd-H-minor-free graphs is a vast generalization of the well-studied
H-minor-free graph families and includes, for example, all
bipartite graphs plus a bounded number of apices. Odd-H-minor-free
graphs are particularly interesting from a structural graph theory
perspective because they break away from the sparsity of H-
minor-free graphs, permitting a quadratic number of edges
Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs
AbstractWe improve the running time of the general algorithmic technique known as Baker’s approach (1994) [1] on H-minor-free graphs from O(nf(|H|)) to O(f(|H|)nO(1)). The numerous applications include, e.g. a 2-approximation for coloring and PTASes for various problems such as dominating set and max-cut, where we obtain similar improvements.On classes of odd-minor-free graphs, which have gained significant attention in recent time, we obtain a similar acceleration for a variant of the structural decomposition theorem proved by Demaine et al. (2010) [20]. We use these algorithms to derive faster 2-approximations; furthermore, we present the first PTASes and subexponential FPT-algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique.We also introduce a technique to derive (nearly) subexponential parameterized algorithms on H-minor-free graphs. Our technique applies, in particular, to problems such as Steiner tree, (directed) subgraph with a property, (directed) longest path, and (connected/independent) dominating set, on some or all proper minor-closed graph classes. We obtain as a corollary that all problems with a minor-monotone subexponential kernel and amenable to our technique can be solved in subexponential FPT-time onH-minor free graphs. This results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
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