Perfect Deletion Research Articles

A Cubic Vertex-Kernel for Trivially Perfect Editing

We consider the Trivially Perfect Editing problem, where one is given an undirected graph \(G = (V,E)\) and a parameter \(k \in {\mathbb {N}}\) and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-complete (Burzyn et al., in Discret Appl Math 154(13):1824–1844, 2006; Nastos and Gao, in Soc Netw 35(3):439–450, 2013) and to admit so-called polynomial kernels (Drange and Pilipczuk, in Algorithmica 80(12):3481–3524, 2018; Guo, in: Tokuyama, (ed) Algorithms and Computation, 18th International Symposium, ISAAC. Lecture Notes in Computer Science, Springer, Sendai, 2007. https://doi.org/10.1007/978-3-540-77120-3_79; Bathie et al., in Algorithmica 1–27, 2022). More precisely, Drange and Pilipczuk (Algorithmica 80(12):3481–3524, 2018) provided \(O(k^7)\) vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem. Notice that a quadratic vertex-kernel was recently obtained for Trivially Perfect Completion by Bathie et al. (Algorithmica 1–27, 2022).

Perfect Multi Deletion Codes Achieve the Asymptotic Optimality of Code Size

This paper studies on the cardinality of perfect multi deletion binary codes. The lower bound for any perfect deletion code with the fixed code length and the number of deletions, and the asymptotic achievable of Levenshtein's upper bound are shown.

A Polynomial Kernel for Trivially Perfect Editing

We give a kernel with O(k^7) vertices for Trivially Perfect Editing, the problem of adding or removing at most k edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao (Soc Netw 35(3):439–450, 2013), and by Liu et al. (Tsinghua Sci Technol 19(4):346–357, 2014). Our general technique implies also the existence of kernels of the same size for related Trivially Perfect Completion and Trivially Perfect Deletion problems. Whereas for the former an O(k^3) kernel was given by Guo (in: ISAAC 2007, LNCS, vol 4835, Springer, pp 915–926, 2007), for the latter no polynomial kernel was known. We complement our study of Trivially Perfect Editing by proving that, contrary to Trivially Perfect Completion, it cannot be solved in time 2^{o(k)}cdot n^{O(1)} unless the exponential time hypothesis fails. In this manner we complete the picture of the parameterized and kernelization complexity of the classic edge modification problems for the class of trivially perfect graphs.

Direct and inverted repeats elicit genetic instability by both exploiting and eluding DNA double-strand break repair systems in mycobacteria.

Repetitive DNA sequences with the potential to form alternative DNA conformations, such as slipped structures and cruciforms, can induce genetic instability by promoting replication errors and by serving as a substrate for DNA repair proteins, which may lead to DNA double-strand breaks (DSBs). However, the contribution of each of the DSB repair pathways, homologous recombination (HR), non-homologous end-joining (NHEJ) and single-strand annealing (SSA), to this sort of genetic instability is not fully understood. Herein, we assessed the genome-wide distribution of repetitive DNA sequences in the Mycobacterium smegmatis, Mycobacterium tuberculosis and Escherichia coli genomes, and determined the types and frequencies of genetic instability induced by direct and inverted repeats, both in the presence and in the absence of HR, NHEJ, and SSA. All three genomes are strongly enriched in direct repeats and modestly enriched in inverted repeats. When using chromosomally integrated constructs in M. smegmatis, direct repeats induced the perfect deletion of their intervening sequences ∼1,000-fold above background. Absence of HR further enhanced these perfect deletions, whereas absence of NHEJ or SSA had no influence, suggesting compromised replication fidelity. In contrast, inverted repeats induced perfect deletions only in the absence of SSA. Both direct and inverted repeats stimulated excision of the constructs from the attB integration sites independently of HR, NHEJ, or SSA. With episomal constructs, direct and inverted repeats triggered DNA instability by activating nucleolytic activity, and absence of the DSB repair pathways (in the order NHEJ>HR>SSA) exacerbated this instability. Thus, direct and inverted repeats may elicit genetic instability in mycobacteria by 1) directly interfering with replication fidelity, 2) stimulating the three main DSB repair pathways, and 3) enticing L5 site-specific recombination.

Parameterized complexity of vertex deletion into perfect graph classes

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F, the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F. Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when parameterized by k, have been stated as open questions in previous work. We show that Perfect Deletion and Weakly Chordal Deletion are W[2]-hard when parameterized by k. In search of positive results, we study a restricted variant of the F-Deletion problem. In this restricted variant, the deleted vertices must be taken from a specified set X, and we parameterize by |X|. We show that for Perfect Deletion and Weakly Chordal Deletion, although this restriction immediately ensures fixed-parameter tractability, it is not enough to yield polynomial kernels, unless NP ⊆ coNP/poly. On the positive side, for Chordal Deletion, the restriction enables us to obtain a kernel with O(|X|4) vertices.

Dynamic Digraph Connectivity Hastens Minimum Sum-of-Diameters Clustering

Dynamic data structures are presented for directed graphs that maintain (a) transitive closure and (b) decomposition into strongly connected components in a "semionline" situation with perfect deletion lookahead but no lookahead for insertions or queries. These algorithms give us "semionline" algorithms for dynamic 2-SAT, as a consequence of which the best known static algorithms for minimum sum-of-diameters clustering are improved by a O(log n) factor.

On the construction of perfect deletion-correcting codes using design theory

We consider a way to construct perfect codes capable of correcting 2 or more deletions using design-theory. As a starting point we use an (ordered) block design to construct a perfect deletion correcting code. Using this code we are able to construct more perfect deletion correcting codes over smaller or larger alphabets by removing or adding symbols in a smart way.

Genetic deletions between directly repeated sequences in bacteriophage T7

DNA sequence analysis of genetic deletions in bacteriophage T7 has shown that these chromosomal rearrangements frequently occur between directly repeated DNA sequences. To study this type of spontaneous deletion in more quantitative detail synthetic fragments of DNA, made by hybridizing two complementary oligonucleotides, were introduced into the non-essential T7 gene 1.3 which codes for T7 DNA ligase. This insert blocked synthesis of functional ligase and made the phage that carried an insert unable to form plaques on a host strain deficient in bacterial ligase. The sequence of the insert was designed so that after it is put into the T7 genome the insert is bracketed by direct repeats. Perfect deletion of the insert between the directly repeated sequences results in a wild-type phage. It was found that these deletion events are highly sensitive to the length of the direct repeats at their ends. In the case of 5 bp direct repeats excision from the genome occurred at a frequency of less than 10(-10), while this value for an almost identical insert bracketed by 10 bp direct repeats was approximately 10(-6). The deletion events were independent of a host recA mutation.