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

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

Graph Concatenation for Quantum Codes

Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code, and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on "graph concatenation", where graphs representing the inner and outer codes are concatenated via a simple graph operation called "generalized local complementation." Our method applies to both binary and non-binary concatenated quantum codes as well as their generalizations.Comment: 26 pages, 12 figures. Figures of concatenated [[5,1,3]] and [[7,1,3]] are added. Submitted to JM

Improved split fluorescent proteins for endogenous protein labeling.

Self-complementing split fluorescent proteins (FPs) have been widely used for protein labeling, visualization of subcellular protein localization, and detection of cell-cell contact. To expand this toolset, we have developed a screening strategy for the direct engineering of self-complementing split FPs. Via this strategy, we have generated a yellow-green split-mNeonGreen21-10/11 that improves the ratio of complemented signal to the background of FP1-10-expressing cells compared to the commonly used split GFP1-10/11; as well as a 10-fold brighter red-colored split-sfCherry21-10/11. Based on split sfCherry2, we have engineered a photoactivatable variant that enables single-molecule localization-based super-resolution microscopy. We have demonstrated dual-color endogenous protein tagging with sfCherry211 and GFP11, revealing that endoplasmic reticulum translocon complex Sec61B has reduced abundance in certain peripheral tubules. These new split FPs not only offer multiple colors for imaging interaction networks of endogenous proteins, but also hold the potential to provide orthogonal handles for biochemical isolation of native protein complexes.Split fluorescent proteins (FPs) have been widely used to visualise proteins in cells. Here the authors develop a screen for engineering new split FPs, and report a yellow-green split-mNeonGreen2 with reduced background, a red split-sfCherry2 for multicolour labeling, and its photoactivatable variant for super-resolution use

Domain interactions within Fzo1 oligomers are essential for mitochondrial fusion

Mitofusins are conserved GTPases essential for the fusion of mitochondria. These mitochondrial outer membrane proteins contain a GTPase domain and two or three regions with hydrophobic heptad repeats, but little is known about how these domains interact to mediate mitochondrial fusion. To address this issue, we have analyzed the yeast mitofusin Fzo1p and find that mutation of any of the three heptad repeat regions (HRN, HR1, and HR2) leads to a null allele. Specific pairs of null alleles show robust complementation, indicating that functional domains need not exist on the same molecule. Biochemical analysis indicates that this complementation is due to Fzo1p oligomerization mediated by multiple domain interactions. Moreover, we find that two non-overlapping protein fragments, one consisting of HRN/GTPase and the other consisting of HR1/HR2, can form a complex that reconstitutes Fzo1p fusion activity. Each of the null alleles disrupts the interaction of these two fragments, suggesting that we have identified a key interaction involving the GTPase domain and heptad repeats essential for fusion

Efficient Solution of Language Equations Using Partitioned Representations

A class of discrete event synthesis problems can be reduced to solving language equations f . X ⊆ S, where F is the fixed component and S the specification. Sequential synthesis deals with FSMs when the automata for F and S are prefix closed, and are naturally represented by multi-level networks with latches. For this special case, we present an efficient computation, using partitioned representations, of the most general prefix-closed solution of the above class of language equations. The transition and the output relations of the FSMs for F and S in their partitioned form are represented by the sets of output and next state functions of the corresponding networks. Experimentally, we show that using partitioned representations is much faster than using monolithic representations, as well as applicable to larger problem instances.Comment: Submitted on behalf of EDAA (http://www.edaa.com/

Pivots, Determinants, and Perfect Matchings of Graphs

We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations.Comment: 16 page