A new approach to graph reconstruction using supercards

The vertex-deleted subgraph G - v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex- deleted subgraphs. The number of common cards of G and H is the cardinality of a maximum multiset of common cards, i.e., the multiset intersection of the decks of G and H. We introduce a new approach to the study of common cards using supercards, where we define a supercard G+ of G and H to be a graph that has at least one vertex-deleted subgraph isomorphic to G, and at least one isomorphic to H. We show how maximum sets of common cards of G and H correspond to certain sets of permutations of the vertices of a supercard, which we call maximum saturating sets. We then show how to construct supercards of various pairs of graphs for which there exists some maximum saturating set X contained in Aut(G+). For certain other pairs of graphs, we show that it is possible to construct G+ and a maximum saturating set X such that the elements of X that are not in Aut(G+) are in one- to-one correspondence with a set of automorphisms of a different supercard G+_lambda � of G and H. Our constructions cover nearly all of the published families of pairs of graphs that have a large number of common cards

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

Size reconstructibility of graphs

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.Comment: 15 page

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

OS diversity for intrusion tolerance: Myth or reality?

One of the key benefits of using intrusion-tolerant systems is the possibility of ensuring correct behavior in the presence of attacks and intrusions. These security gains are directly dependent on the components exhibiting failure diversity. To what extent failure diversity is observed in practical deployment depends on how diverse are the components that constitute the system. In this paper we present a study with operating systems (OS) vulnerability data from the NIST National Vulnerability Database. We have analyzed the vulnerabilities of 11 different OSes over a period of roughly 15 years, to check how many of these vulnerabilities occur in more than one OS. We found this number to be low for several combinations of OSes. Hence, our analysis provides a strong indication that building a system with diverse OSes may be a useful technique to improve its intrusion tolerance capabilities

On rainbow tetrahedra in Cayley graphs

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations. Generalizing asymptotic results are discussed as well.Comment: 21 pages, 7 figure

Cutoff for non-backtracking random walks on sparse random graphs

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape

A computer system to perform structure comparison using TOPS representations of protein structure

We describe the design and implementation of a fast topology–based method for protein structure comparison. The approach uses the TOPS topological representation of protein structure, aligning two structures using a common discovered pattern and generating measure of distance derived from an insert score. Heavy use is made of a constraint-based pattern matching algorithm for TOPS diagrams that we have designed and described elsewhere Gilbert et al. (1999). The comparison system is maintained at the European Bioinformatics Institute and is available over the Web via the at tops.ebi.ac.uk/tops. Users submit a structure description in Protein Data Bank (PDB) format and can compare it with structures in the entire PDB or a representative subset of protein domains, receiving the results by email