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Bhalchandra D. Thatte

Publications and source records attributed to Bhalchandra D. Thatte.

At least 19 recordsLinked to original sources
arXiv2024

On the class reconstruction number of trees

Harary and Lauri conjectured that the class reconstruction number of trees is 2, that is, each tree has two unlabelled vertex-deleted subtrees that are not both in the deck of any other tree. We show that each tree $T$ can be reconstructed up to isomorphism given two of its unlabelled subgraphs $T-u$ and $T-v$ under the assumption that $u$ and $v$ are chosen in a particular way. Our result does not completely resolve the conjecture of Harary and Lauri since the special property defining $u$ and $v$ cannot be recognised from the given subtrees $T-u$ and $T-v$.

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arXiv2023

A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs

Kelly's lemma is a basic result on graph reconstruction. It states that given the deck of a graph $G$ on $n$ vertices, and a graph $F$ on fewer than $n$ vertices, we can count the number of subgraphs of $G$ that are isomorphic to $F$. Moreover, for a given card $G-v$ in the deck, we can count the number of subgraphs of $G$ that are isomorphic to $F$ and that contain $v$. We consider the problem of refining the lemma to count rooted subgraphs such that the root vertex coincides the deleted vertex. We show that such counting is not possible in general, but a multiset of rooted subgraphs of a fixed height $k$ can be counted if $G$ has radius more than $k$. We also prove a similar result for the edge reconstruction problem.

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arXiv2022

On the Boolean dimension of a graph and other related parameters

We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.

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arXiv2020

A construction of the abstract induced subgraph poset of a graph from its abstract edge subgraph poset

The abstract induced subgraph poset of a graph is the isomorphism class of the induced subgraph poset of the graph, suitably weighted by subgraph counting numbers. The abstract bond lattice and the abstract edge-subgraph poset are defined similarly by considering the lattice of subgraphs induced by connected partitions and the poset of edge-subgraphs, respectively. Continuing our development of graph reconstruction theory on these structures, we show that if a graph has no isolated vertices, then its abstract bond lattice and the abstract induced subgraph poset can be constructed from the abstract edge-subgraph poset except for the families of graphs that we characterise. The construction of the abstract induced subgraph poset from the abstract edge-subgraph poset generalises a well known result in reconstruction theory that states that the vertex deck of a graph with at least 4 edges and without isolated vertices can be constructed from its edge deck.12

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arXiv2015

Subgraph posets and graph reconstruction

We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconstruction questions about the chromatic symmetric function and the symmetric Tutte polynomial. In particular, we show that the symmetric Tutte polynomial of a tree can be constructed from its chromatic symmetric function. We classify graphs that are not reconstructible from their abstract edge-subgraph posets, and further show that the families presented here are the only graphs not Q-reconstructible if and only if the edge reconstruction conjecture is true. Let f be a bijection from the set of all unlabelled graphs to itself such that for all unlabelled graphs G and H, hom(G,H) = hom(f(G), f(H)). We conjecture that f is an identity map. We show that this conjecture is weaker than the edge reconstruction conjecture. Our conjecture is motivated by homomorphism cancellation results due to Lovász.

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arXiv2014

An algebraic formulation of the graph reconstruction conjecture

The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $ψ(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $ψ(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $ψ(n) = d(n)$. We give a framework based on Kocay's lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) \geq \mathsf{rank}_\mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $ψ(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = \mathsf{rank}_\mathbb{R}(M)$.

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arXiv2013

The maximum agreement subtree problem

In this paper we investigate an extremal problem on binary phylogenetic trees. Given two such trees $T_1$ and $T_2$, both with leaf-set ${1,2,...,n}$, we are interested in the size of the largest subset $S \subseteq {1,2,...,n}$ of leaves in a common subtree of $T_1$ and $T_2$. We show that any two binary phylogenetic trees have a common subtree on $Ω(\sqrt{\log{n}})$ leaves, thus improving on the previously known bound of $Ω(\log\log n)$ due to M. Steel and L. Szekely. To achieve this improved bound, we first consider two special cases of the problem: when one of the trees is balanced or a caterpillar, we show that the largest common subtree has $Ω(\log n)$ leaves. We then handle the general case by proving and applying a Ramsey-type result: that every binary tree contains either a large balanced subtree or a large caterpillar. We also show that there are constants $c, α> 0$ such that, when both trees are balanced, they have a common subtree on $c n^α$ leaves. We conjecture that it is possible to take $α= 1/2$ in the unrooted case, and both $c = 1$ and $α= 1/2$ in the rooted case.

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arXiv2011

Reconstructing pedigrees: some identifiability questions for a recombination-mutation model

Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees - Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.

q-bio.PE↗
arXiv2009

Revisiting an equivalence between maximum parsimony and maximum likelihood methods in phylogenetics

Tuffley and Steel (1997) proved that Maximum Likelihood and Maximum Parsimony methods in phylogenetics are equivalent for sequences of characters under a simple symmetric model of substitution with no common mechanism. This result has been widely cited ever since. We show that small changes to the model assumptions suffice to make the two methods inequivalent. In particular, we analyze the case of bounded substitution probabilities as well as the molecular clock assumption. We show that in these cases, even under no common mechanism, Maximum Parsimony and Maximum Likelihood might make conflicting choices. We also show that if there is an upper bound on the substitution probabilities which is `sufficiently small', every Maximum Likelihood tree is also a Maximum Parsimony tree (but not vice versa).

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arXiv2009

Maximum Parsimony on Subsets of Taxa

In this paper we investigate mathematical questions concerning the reliability (reconstruction accuracy) of Fitch's maximum parsimony algorithm for reconstructing the ancestral state given a phylogenetic tree and a character. In particular, we consider the question whether the maximum parsimony method applied to a subset of taxa can reconstruct the ancestral state of the root more accurately than when applied to all taxa, and we give an example showing that this indeed is possible. A surprising feature of our example is that ignoring a taxon closer to the root improves the reliability of the method. On the other hand, in the case of the two-state symmetric substitution model, we answer affirmatively a conjecture of Li, Steel and Zhang which states that under a molecular clock the probability that the state at a single taxon is a correct guess of the ancestral state is a lower bound on the reconstruction accuracy of Fitch's method applied to all taxa.

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arXiv2007

Reconstructing pedigrees: a stochastic perspective

A pedigree is a directed graph that describes how individuals are related through ancestry in a sexually-reproducing population. In this paper we explore the question of whether one can reconstruct a pedigree by just observing sequence data for present day individuals. This is motivated by the increasing availability of genomic sequences, but in this paper we take a more theoretical approach and consider what models of sequence evolution might allow pedigree reconstruction (given sufficiently long sequences). Our results complement recent work that showed that pedigree reconstruction may be fundamentally impossible if one uses just the degrees of relatedness between different extant individuals. We find that for certain stochastic processes, pedigrees can be recovered up to isomorphism from sufficiently long sequences.

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arXiv2006

Combinatorics of pedigrees

A pedigree is a directed graph in which each vertex (except the founder vertices) has two parents. The main result in this paper is a construction of an infinite family of counter examples to a reconstruction problem on pedigrees, thus negatively answering a question of Steel and Hein. Some positive reconstruction results are also presented. The problem of counting distinct (mutually non-isomorphic) pedigrees is considered. The known lower and upper bounds on the number of pedigrees are improved upon, and their relevance to pedigree reconstruction from DNA sequence data is discussed. It is shown that the information theoretic bound on the number of segregating sites in the sequence data that is minimally essential for reconstructing pedigrees would not significantly change with improved enumerative estimates.

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arXiv2006

Invertibility of the TKF model of sequence evolution

We consider character sequences evolving on a phylogenetic tree under the TKF91 model. We show that as the sequence lengths tend to infinity the the topology of the phylogenetic tree and the edge lengths are determined by any one of (a) the alignment of sequences (b) the collection of sequence lengths. We also show that the probability of any homology structure on a collection of sequences related by a TKF91 process on a tree is independent of the root location. Keywords: phylogenetics, DNA sequence evolution models, identifiability, alignment

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arXiv2006

$G$-reconstruction of graphs

Let $G$ be a group of permutations acting on an $n$-vertex set $V$, and $X$ and $Y$ be two simple graphs on $V$. We say that $X$ and $Y$ are $G$-isomorphic if $Y$ belongs to the orbit of $X$ under the action of $G$. One can naturally generalize the reconstruction problems so that when $G$ is $S_n$, the symmetric group, we have the usual reconstruction problems. In this paper, we study $G$-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.

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arXiv2006

A reconstruction problem related to balance equations-I

A modified $k$-deck of a graph is obtained by removing $k$ edges in all possible ways and adding $k$ (not necessarily new) edges in all possible ways. Krasikov and Roditty used these decks to give an independent proof of Müller's result on the edge reconstructibility of graphs. They asked if a $k$-edge deck could be constructed from its modified $k$-deck. In this paper, we solve the problem when $k=1$. We also offer new proofs of Lovász's result, one describing the constructed graph explicitly, (thus answering a question of Bondy), and another based on the eigenvalues of Johnson graph.

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arXiv2005

A reconstruction problem related to balance equations-II: the general case

A modified $k$-deck of a graph $G$ is obtained by removing $k$ edges of $G$ in all possible ways, and adding $k$ (not necessarily new) edges in all possible ways. Krasikov and Roditty asked if it was possible to construct the usual $k$-edge deck of a graph from its modified $k$-deck. Earlier I solved this problem for the case when $k=1$. In this paper, the problem is completely solved for arbitrary $k$. The proof makes use of the $k$-edge version of Lovász's result and the eigenvalues of certain matrix related to the Johnson graph. This version differs from the published version. Lemma 2.3 in the published version had a typo in one equation. Also, a long manipulation of some combinatorial expressions was skipped in the original proof of Lemma 2.3, which made it difficult to follow the proof. Here a clearer proof is given.

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