92,708 research outputs found

### The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

We study the interplay between principal pivot transform (pivot) and loop
complementation for graphs. This is done by generalizing loop complementation
(in addition to pivot) to set systems. We show that the operations together,
when restricted to single vertices, form the permutation group S_3. This leads,
e.g., to a normal form for sequences of pivots and loop complementation on
graphs. The results have consequences for the operations of local
complementation and edge complementation on simple graphs: an alternative proof
of a classic result involving local and edge complementation is obtained, and
the effect of sequences of local complementations on simple graphs is
characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and
Remark 2

### Hypomorphy of graphs up to complementation

Let $V$ be a set of cardinality $v$ (possibly infinite). Two graphs $G$ and
$G'$ with vertex set $V$ are {\it isomorphic up to complementation} if $G'$ is
isomorphic to $G$ or to the complement $\bar G$ of $G$. Let $k$ be a
non-negative integer, $G$ and $G'$ are {\it $k$-hypomorphic up to
complementation} if for every $k$-element subset $K$ of $V$, the induced
subgraphs $G\_{\restriction K}$ and $G'\_{\restriction K}$ are isomorphic up to
complementation. A graph $G$ is {\it $k$-reconstructible up to complementation}
if every graph $G'$ which is $k$-hypomorphic to $G$ up to complementation is in
fact isomorphic to $G$ up to complementation. We give a partial
characterisation of the set $\mathcal S$ of pairs $(n,k)$ such that two graphs
$G$ and $G'$ on the same set of $n$ vertices are equal up to complementation
whenever they are $k$-hypomorphic up to complementation. We prove in particular
that $\mathcal S$ contains all pairs $(n,k)$ such that $4\leq k\leq n-4$. We
also prove that 4 is the least integer $k$ such that every graph $G$ having a
large number $n$ of vertices is $k$-reconstructible up to complementation; this
answers a question raised by P. Ill

### Complementation, Local Complementation, and Switching in Binary Matroids

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex
set $V$ can be obtained from a complete graph on $V$ via a sequence of the
operations of complementation, switching edges and non-edges at a vertex, and
local complementation. The last operation involves taking the complement in the
neighbourhood of a vertex. In this paper, we consider natural generalizations
of these operations for binary matroids and explore their behaviour. We
characterize all binary matroids obtainable from the binary projective geometry
of rank $r$ under the operations of complementation and switching. Moreover, we
show that not all binary matroids of rank at most $r$ can be obtained from a
projective geometry of rank $r$ via a sequence of the three generalized
operations. We introduce a fourth operation and show that, with this additional
operation, we are able to obtain all binary matroids.Comment: Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math.
(2020

### Can Nondeterminism Help Complementation?

Complementation and determinization are two fundamental notions in automata
theory. The close relationship between the two has been well observed in the
literature. In the case of nondeterministic finite automata on finite words
(NFA), complementation and determinization have the same state complexity,
namely Theta(2^n) where n is the state size. The same similarity between
determinization and complementation was found for Buchi automata, where both
operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing
question is whether there exists a type of omega-automata whose determinization
is considerably harder than its complementation. In this paper, we show that
for all common types of omega-automata, the determinization problem has the
same state complexity as the corresponding complementation problem at the
granularity of 2^\Theta(.).Comment: In Proceedings GandALF 2012, arXiv:1210.202

### On self-complementation

We prove that, with very few exceptions, every graph of order n, n - 0, 1(mod 4) and size at most n - 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs

### Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique

In this paper, we first introduce a lower bound technique for the state
complexity of transformations of automata. Namely we suggest first considering
the class of full automata in lower bound analysis, and later reducing the size
of the large alphabet via alphabet substitutions. Then we apply such technique
to the complementation of nondeterministic \omega-automata, and obtain several
lower bound results. Particularly, we prove an \omega((0.76n)^n) lower bound
for B\"uchi complementation, which also holds for almost every complementation
or determinization transformation of nondeterministic omega-automata, and prove
an optimal (\omega(nk))^n lower bound for the complementation of generalized
B\"uchi automata, which holds for Streett automata as well

### Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer
$h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that
two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set
$V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever
${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}.
Let $s(h)$ be the least integer $k$ such that the conclusion above holds and
let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)=
h+2^{\lfloor \log_2 h\rfloor}$. In the special case $h=2^{\ell}$ or
$h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and
$v(2)=6$ were obtained in a previous work.Comment: 15 page

### Tight Upper Bounds for Streett and Parity Complementation

Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound $2^{\Omega(n\lg nk)}$ and upper
bound $2^{O(nk\lg nk)}$, where $n$ is the state size, $k$ is the number of
Streett pairs, and $k$ can be as large as $2^{n}$. Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound $2^{O(n \lg
n+nk \lg k)}$ for $k = O(n)$ and $2^{O(n^{2} \lg n)}$ for $k = \omega(n)$,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound $2^{O(n \lg n)}$ for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011

### State of B\"uchi Complementation

Complementation of B\"uchi automata has been studied for over five decades
since the formalism was introduced in 1960. Known complementation constructions
can be classified into Ramsey-based, determinization-based, rank-based, and
slice-based approaches. Regarding the performance of these approaches, there
have been several complexity analyses but very few experimental results. What
especially lacks is a comparative experiment on all of the four approaches to
see how they perform in practice. In this paper, we review the four approaches,
propose several optimization heuristics, and perform comparative
experimentation on four representative constructions that are considered the
most efficient in each approach. The experimental results show that (1) the
determinization-based Safra-Piterman construction outperforms the other three
in producing smaller complements and finishing more tasks in the allocated time
and (2) the proposed heuristics substantially improve the Safra-Piterman and
the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in
the Proceedings of the 15th International Conference on Implementation and
Application of Automata (CIAA

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