# Bhalchandra D. Thatte's research while affiliated with Federal University of Minas Gerais and other places

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## Publications (30)

We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions and 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.

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, respectiv...

Previously we showed that many invariants of a graph can be computed from its abstract induced subgraph poset, which is the isomorphism class of the induced subgraph poset, suitably weighted by subgraph counting numbers. In this paper, we study the abstract bond lattice of a graph, which is the isomorphism class of the lattice of distinct unlabelle...

We consider only finite simple graphs in this paper. Earlier we showed that many invariants of a graph can be computed from the isomorphism class of its partially ordered set of distinct unlabeled non‐empty induced subgraphs, that is, the subgraphs themselves are not required. In this paper, we consider an analogous problem of reconstructing an arb...

In this paper we present a distributed algorithm for detecting cycles in large-scale directed graphs, along with its correctness proof and analysis. The algorithm is then extended to find strong components in directed graphs. We indicate an application to detecting cycles in number theoretic functions such as the proper divisor function. Our protot...

We derive the exact one-step transition probabilities of the number of lineages that are ancestral to a random sample from the current generation of a bi-parental population that is evolving under the discrete Wright–Fisher model with \(n\) diploid individuals. Our model allows for a per-generation recombination probability of \(r\). When \(r=1\),...

The graph reconstruction conjecture asserts that a finite simple graph on at
least 3 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...

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...

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 id...

We consider a rooted phylogenetic tree under molecular clock, a two-state charac- ter, and a two-state symmetric substitution model (Neyman model). We examine the problem of inferring the ancestral root state by the maximum likelihood and the maximum parsimony methods. In particular, we would like to investigate if there are trees and characters fo...

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...

Tuffley and Steel (Bull. Math. Biol. 59:581-607, 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 suf...

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 th...

A pedigree is a finite directed acyclic graph in which all vertices other than the founder vertices have two outgoing arcs each (to their parents), while the founder vertices have no outgoing arcs. The primary motivation to study pedigrees comes from pop- ulation biology. The main result in this paper is a construction of an infinite family of coun...

This paper deals with crtain posets and lattices associated with a graph. In my earlier paper I showed that several invariants of a graph can be computed from the isomorphism class of its poset of non-empty induced subgraphs. In this paper I will prove that the (abstract and folded) connected partition lattice of a graph can be constructed from its...

We give upper bounds on the order of the automorphism group of a simple graph

A claw is an induced subgraph isomorphic to K1,3. The claw-point is the point of degree 3 in a claw. A graph is called p-claw-free when no p-cycle has a claw-point on it. It is proved that for p ≥ 4, p-claw-free graphs containting at least one chordless p-cycle are edge reconstructible. It is also proved that chordal graphs are edge reconstructible...

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 pr...

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 reconstruct...

McMorris and Powers proved an Arrow-type theorem on phylogenies given as collections of quartets. There is an error in one of the main lemmas used to prove this theorem. However, this lemma (and thereby the theorem) is still true, and a correct proof is provided.

We consider character sequences evolving on a phylogenetic tree under the TKF91 model [J. Thorne, H. Kishino and J. Felsenstein, An evolutionary model for maximum likelihood alignment of DNA sequences. J. Mol. Evol. 33, 114 ff (1991)]. We show that as the sequence lengths tend to infinity the topology of the phylogenetic tree and the edge lengths a...

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 pa...

Given a graph G, an incidence matrix N(G) is defined for the set of distinct isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then every graph invariant must be reconstructible from this matrix, even when the graphs indexing the rows and the columns of N(G) are unspecified. It is proved that the characteristic polynomial,...

Abstract We give upper bounds,on the order of the automorphism,group of a simple graph In this note we present some,upper bounds,on the order of the automorphism,group of a graph, which is assumed to be simple, having no loops or multiple edges. Somewhat surprisingly, we did not find such bounds in the literature and the goal of this paper is to fi...

We give upper bounds on the order of the automorphism group of a simple graph

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,...

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 reconstructability of graphs. They asked if a k-edge deck could be constructed from its modified k-deck. In...

This note supplements an earlier paper of this author, in which the concept of a strong k-hypomorphism between two graphs was defined (Thatte, 1990, Sectin VI). For k=1, this is just a hypomorphism. Here it is proved that strongly k-hypomorphic graphs and strongly k-edge hypomorphic directed graphs are isomorphic if k>1.

Tutte (1979) proved that the disconnected spanning subgraphs of a graph can be reconstructed from its vertex deck. This result is used to prove that if we can reconstruct a set of connected graphs from the shuffled edge deck (SED) then the vertex reconstruction conjecture is true. It is proved that a set of connected graphs can be reconstructed fro...

A generalization of Nash-Williams’ lemma,is proved for the structure of m-uniform null (m − k)-designs. It is then applied to various graph reconstruction problems. A short combinatorial proof of the edge reconstructibility of digraphs having regular underlying undirected graphs (e.g. tournaments) is given. A type of Nash-Williams’ lemma is conject...

## Citations

... The bond lattice L G of G is the set of all connected partitions of G, partially ordered by refinement. It is well-known that L G is a geometric lattice with the rank of each π being given by |V (G)| − |π|, where |π| denotes the number of blocks of π, see [27, Theorem 1.1 and 2.6] and also [29]. The Tutte polynomial of G is the polynomial defined by ...

... Finding cyclic structures in a graph has been largely studied in the literature because of its many potential application scenarios. Some existing works have been interested in detecting circuits of directed or undirected graphs [16][17][18][19]. Other studies focused on finding the shortest circuit [20][21][22] or the longest circuits of a graph [23][24][25][26]. ...

... Another reconstructible graph invariants are characteristic polynomial [30], chromatic polynomial [30] and planarity [1]. Oliveira and Thatte [26,29] had a new approach to this problem by studying the matrix of covering numbers of graphs by sequences of subgraphs and proposing a bound for the rank of this matrix. ...

... Theorem 6 [16]. Graphs having no vertex lying on more than one induced P 4 are reconstructible. ...

... The theoretical properties of maximum parsimony for ancestral state reconstruction have been widely studied [34,36,12,37,13,6,14,16,17,18]. From its very definition, one might expect maximum parsimony to perform well when the probabilities of substitution along the edges are "sufficiently small." ...

... Finally, we note that other extensions to tree-based Markov models are possible. For instance, adapting a model used in pedigree reconstruction (Thatte 2012), Francis and Moulton (2018) introduced an alternative probabilistic recombination-mutation model and established identifiability for almost the entire class of tree-child networks under this model (and under the mild assumption that the root of the network is not the parent of a reticulation vertex). We refer the reader to Francis and Moulton (2018) for further details on the model assumptions. ...

... It was an open question until recently how small MAST( 1 , 2 ) could be in terms of the number of leaves . In his recent paper [6], Markin established that the minimum is in fact of order Θ(log ) by improving the previous lower bound that was of order √log [7]. ; see [2]. ...

Reference: Trees in Many Contexts

... Moreover, it has recently been shown that rooted phylogenetic networks also cannot be reconstructed uniquely from their subnetworks obtained by deleting one or more leaves and transforming the result into a valid rooted phylogenetic network (Huber et al. 2014). A similar reconstruction question for pedigrees has also been answered negatively (Thatte 2008). ...

... Theoretical aspects of the size of automorphism groups of simple graphs are discussed by Krasikov. [15] An atom (vertex) orbit is the set of constitutionally equivalent atoms. The column for atom 1 in Table 1 shows that atom 1 can be replaced by atoms 3 or 4, consequently atoms 1, 3 and 4 form an atom orbit (given in the lower part of Table 1, denoted by "Sets"). ...

Reference: Automorphism Groups of Alkane Graphs

... An important consequence of Conjecture 2 would be that chordal graphs are edge and vertex reconstructable. The edge-reconstructability has already been proved by B.D. That- te [19]. Finally in Section 4 we derive in Theorem 9 a new formula for the colored Jones function. ...