92,708 research outputs found

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

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    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

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    Let VV be a set of cardinality vv (possibly infinite). Two graphs GG and G′G' with vertex set VV are {\it isomorphic up to complementation} if G′G' is isomorphic to GG or to the complement Gˉ\bar G of GG. Let kk be a non-negative integer, GG and G′G' are {\it kk-hypomorphic up to complementation} if for every kk-element subset KK of VV, the induced subgraphs G_↾KG\_{\restriction K} and G′_↾KG'\_{\restriction K} are isomorphic up to complementation. A graph GG is {\it kk-reconstructible up to complementation} if every graph G′G' which is kk-hypomorphic to GG up to complementation is in fact isomorphic to GG up to complementation. We give a partial characterisation of the set S\mathcal S of pairs (n,k)(n,k) such that two graphs GG and G′G' on the same set of nn vertices are equal up to complementation whenever they are kk-hypomorphic up to complementation. We prove in particular that S\mathcal S contains all pairs (n,k)(n,k) such that 4≤k≤n−44\leq k\leq n-4. We also prove that 4 is the least integer kk such that every graph GG having a large number nn of vertices is kk-reconstructible up to complementation; this answers a question raised by P. Ill

    Complementation, Local Complementation, and Switching in Binary Matroids

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    In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set VV can be obtained from a complete graph on VV 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 rr under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most rr can be obtained from a projective geometry of rank rr 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?

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    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

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    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

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    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

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

    Tight Upper Bounds for Streett and Parity Complementation

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    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 2Ω(nlg⁡nk)2^{\Omega(n\lg nk)} and upper bound 2O(nklg⁡nk)2^{O(nk\lg nk)}, where nn is the state size, kk is the number of Streett pairs, and kk can be as large as 2n2^{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 2O(nlg⁡n+nklg⁡k)2^{O(n \lg n+nk \lg k)} for k=O(n)k = O(n) and 2O(n2lg⁡n)2^{O(n^{2} \lg n)} for k=ω(n)k = \omega(n), which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a tight upper bound 2O(nlg⁡n)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

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    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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