6,749 research outputs found
Reset thresholds of automata with two cycle lengths
We present several series of synchronizing automata with multiple parameters,
generalizing previously known results. Let p and q be two arbitrary co-prime
positive integers, q > p. We describe reset thresholds of the colorings of
primitive digraphs with exactly one cycle of length p and one cycle of length
q. Also, we study reset thresholds of the colorings of primitive digraphs with
exactly one cycle of length q and two cycles of length p.Comment: 11 pages, 5 figures, submitted to CIAA 201
Counting Proper Mergings of Chains and Antichains
A proper merging of two disjoint quasi-ordered sets and is a
quasi-order on the union of and such that the restriction to and
yields the original quasi-order again and such that no elements of and
are identified. In this article, we consider the cases where and
are chains, where and are antichains, and where is an antichain and
is a chain. We give formulas that determine the number of proper mergings
in all three cases, and introduce two new bijections from proper mergings of
two chains to plane partitions and from proper mergings of an antichain and a
chain to monotone colorings of complete bipartite digraphs. Additionally, we
use these bijections to count the Galois connections between two chains, and
between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
New Results on Directed Edge Dominating Set
We study a family of generalizations of Edge Dominating Set on directed
graphs called Directed -Edge Dominating Set. In this problem an arc
is said to dominate itself, as well as all arcs which are at distance
at most from , or at distance at most to .
First, we give significantly improved FPT algorithms for the two most
important cases of the problem, -dEDS and -dEDS (that correspond
to versions of Dominating Set on line graphs), as well as polynomial kernels.
We also improve the best-known approximation for these cases from logarithmic
to constant. In addition, we show that -dEDS is FPT parameterized by
, but W-hard parameterized by (even if the size of the optimal is
added as a second parameter), where is the treewidth of the underlying
graph of the input.
We then go on to focus on the complexity of the problem on tournaments. Here,
we provide a complete classification for every possible fixed value of ,
which shows that the problem exhibits a surprising behavior, including cases
which are in P; cases which are solvable in quasi-polynomial time but not in P;
and a single case which is NP-hard (under randomized reductions) and
cannot be solved in sub-exponential time, under standard assumptions
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