105,147 research outputs found
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
Packing spanning graphs from separable families
Let be a separable family of graphs. Then for all positive
constants and and for every sufficiently large integer ,
every sequence of graphs of order and maximum
degree at most such that packs into . This improves results of
B\"ottcher, Hladk\'y, Piguet, and Taraz when is the class of trees
and of Messuti, R\"odl, and Schacht in the case of a general separable family.
The result also implies approximate versions of the Oberwolfach problem and of
the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees
have maximum degree at most . The proof uses the local resilience of
random graphs and a special multi-stage packing procedure
Independent sets of maximum weight in apple-free graphs
We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free
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