Stronger reconstruction of distance-hereditary graphs

A graph is said to be set-reconstructible if it is uniquely determined up to isomorphism from the set S of its non-isomorphic one-vertex deleted unlabeled subgraphs. Harary’s conjecture asserts that every finite simple undirected graph on four or more vertices is set-reconstructible. A graph G is said to be distance-hereditary if for all connected induced subgraph F of G, dF (u, v) = dG(u, v) for every pair of vertices u, v ∈ V (F). In this paper, we have proved that the class of all 2-connected distance-hereditary graphs G with diam(G) = 2 or diam(G) = diam(Ḡ) = 3 are set-reconstructible.The second author is supported by the University Grants Commission, Government of India. (F./2017-18/NFO-2017-18-OBC-TAM-53159)Publisher's Versio

On some problems in reconstruction

A graph is {\it reconstructible} if it is determined by its {\it deck} of unlabeled subgraphs obtained by deleting one vertex; a {\it card} is one of these subgraphs. The {\it Reconstruction Conjecture} asserts that all graphs with at least three vertices are reconstructible. In Chapter $2$ we consider $k$-deck reconstruction of graphs. The {\it $k$-deck} of a graph is its multiset of $k$-vertex induced subgraphs. We prove a generalization of a result by Bollob\'as concerning the $k$-deck reconstruction of almost all graphs, showing that when $\ell \le (1-\epsilon)\frac{n}{2}$, the probability than an $n$-vertex graph is reconstructible from some $\binom{\ell+1}{2}$ of the graphs in the $(n-\ell)$-deck tends to $1$ as $n$ tends to $\infty$. We determine the smallest $k$ such that all graphs with maximum degree $2$ are $k$-deck reconstructible. We prove for $n\ge 26$ that whether a graph is connected is determined by its $(n-3)$-deck. We prove that if $G$ is a complete $r$-partite graphs, then $G$ is $(r+1)$-deck reconstructible (the same holds for $\overline{G}$). In Chapter $3$ we consider degree-associated reconstruction. An $(n-1)$-vertex induced subgraph accompanied with the degree of the missing vertex is called a {\it dacard}. The {\it degree-associated reconstruction number} of a graph $G$ is the fewest number of dacards needed to determine $G$. We provide a tool for reconstructing some graphs from two dacards. We prove that certain families of trees and disconnected graphs can be reconstructed from two dacards. We also determine the degree-associated reconstruction number for complete multipartite graphs and their complements. For such graphs, we also determine the least $s$ such that {\it every} set of $s$ dacards determine the graph. In Chapter $4$ we consider the reconstruction of matrices from principal submatrices. A $(n-\ell)$-by-$(n-\ell)$ principal submatrix is a submatrix formed by deleting $\ell$ rows and columns symmetrically. The {\it matrix reconstruction threshold} $mrt(\ell)$ is the minimum integer $n_0$ such that for $n\ge n_0$ all $n$-by-$n$ matrices are reconstructible from their deck of $(n-\ell)$-by-$(n-\ell)$ principal submatrices. We prove $mrt(\ell) \leq \frac{2}{\ln 2}\ell^2+3\ell$

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 $\psi(n)$ 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 $\psi(n) - d(n)$ 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 $\psi(n) = d(n)$. 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 $\psi(n)$. 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 $d(n) = \mathsf{rank}_\mathbb{R}(M)$.Comment: 12 pages, 2 figure

Isomorph-free generation of 2-connected graphs with applications

Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Lossless Representation of Graphs using Distributions

We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their sub-triangles. In the second part, we propose graph representations in terms of one-dimensional distributions (e.g., distribution of the node weights, sum of adjacent weights, etc.). For the case when the weights of the graph are real-valued vectors, we show that all graphs, except for a set of measure zero, are uniquely determined, up to isomorphism, from these distributions. The motivating application for this paper is the problem of browsing through large sets of graphs.Comment: 19 page