13,882 research outputs found
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
A Survey of Best Monotone Degree Conditions for Graph Properties
We survey sufficient degree conditions, for a variety of graph properties,
that are best possible in the same sense that Chvatal's well-known degree
condition for hamiltonicity is best possible.Comment: 25 page
Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian
This note shows there are infinitely many finite groups G, such that every
connected Cayley graph on G has a hamiltonian cycle, and G is not solvable.
Specifically, for every prime p that is congruent to 1, modulo 30, we show
there is a hamiltonian cycle in every connected Cayley graph on the direct
product of the cyclic group of order p with the alternating group A_5 on five
letters.Comment: 7 pages, plus a 22-page appendix of notes to aid the refere
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