3,227 research outputs found
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
Hereditary chip-firing models generalize the Abelian sandpile model and the
cluster firing model to an exponential family of games induced by covers of the
vertex set. This generalization retains some desirable properties, e.g.
stabilization is independent of firings chosen and each chip-firing equivalence
class contains a unique recurrent configuration. In this paper we present an
explicit bijection between the recurrent configurations of a hereditary
chip-firing model on a graph and its spanning trees.Comment: 13 page
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
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
Complexity of C_k-Coloring in Hereditary Classes of Graphs
For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs.
We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw
Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull
The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex
weights asks for a set of pairwise nonadjacent vertices of maximum total
weight. Being one of the most investigated and most important problems on
graphs, it is well known to be NP-complete and hard to approximate. The
complexity of MWIS is open for hole-free graphs (i.e., graphs without induced
subgraphs isomorphic to a chordless cycle of length at least five). By applying
clique separator decomposition as well as modular decomposition, we obtain
polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as
for odd-hole- and bull-free graphs (dart and bull have five vertices, say
, and dart has edges , while bull has edges
). If the graphs are hole-free instead of odd-hole-free then
stronger structural results and better time bounds are obtained
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